Glossary

Short definitions of the terms this book uses, from A to Z, each with a link to the chapter that teaches it.

A

Abel-Ruffini theorem There is no formula in radicals for the roots of the general polynomial of degree 5 or more. Ruffini gave an incomplete proof in 1799 and Abel a complete one in 1824; Galois theory explains why. See Chapter 20.

Abelian group A group in which \(ab = ba\) for all elements \(a\) and \(b\), named after Niels Henrik Abel. Abelian groups are also called commutative, and are often written with \(+\). See Chapter 13.

Abelianization The quotient \(G/[G, G]\) of a group by the subgroup generated by its commutators: the largest abelian quotient of \(G\). It is an example of a functor from groups to abelian groups. See Chapter 22.

Absolute value The size of a number without its sign: \(|a| = a\) if \(a \ge 0\) and \(|a| = -a\) if \(a < 0\). The number \(|a - b|\) is the distance between \(a\) and \(b\) on the number line. See Chapter 2.

Action (of a group) A rule that gives, for each \(g\) in a group \(G\) and each \(x\) in a set \(X\), a point \(g \cdot x\) of \(X\), such that the identity moves nothing and \((gh) \cdot x = g \cdot (h \cdot x)\). The set \(X\) is then called a \(G\)-set. See Chapter 16.

Affine variety The set of common zeros in \(k^n\) of a set of polynomials in \(k[x_1, \ldots, x_n]\). Circles, parabolas and the twisted cubic are examples. See Chapter 22.

Algebraic closure An algebraic extension of a field that is algebraically closed. Every field has one, unique up to isomorphism; the algebraic closure of \(\mathbb{R}\) is \(\mathbb{C}\), and that of \(\mathbb{Q}\) is the field of algebraic numbers. See Chapter 19.

Algebraic element An element \(\alpha\) of an extension \(K\) of \(F\) is algebraic over \(F\) if \(f(\alpha) = 0\) for some nonzero polynomial \(f\) with coefficients in \(F\). An extension is algebraic if all its elements are. See Chapter 19.

Algebraic integer A complex number that is a root of a monic polynomial with integer coefficients, such as \(\sqrt2\) or \(i\). The algebraic integers in a number field form its ring of integers. See Chapter 22.

Algebraic number A complex number that is a root of a nonzero polynomial with rational coefficients. The algebraic numbers form an algebraically closed field, \(\overline{\mathbb{Q}}\). See Chapter 12.

Algebraically closed A field in which every nonconstant polynomial has a root, and so splits into linear factors. The fundamental theorem of algebra says that \(\mathbb{C}\) is algebraically closed. See Chapter 12.

Alternating group The group \(A_n\) of even permutations of \(\{1, \ldots, n\}\). It has \(n!/2\) elements for \(n \ge 2\), and it is simple for \(n \ge 5\). See Chapter 13.

Analytic geometry The study of shapes through equations in coordinates, so that geometric questions become algebra and equations can be seen as pictures. It was found independently by Descartes and Fermat. See Chapter 3.

Argument (of a complex number) An angle \(\theta\) with \(z = r(\cos\theta + i\sin\theta)\), written \(\arg z\). It is fixed only up to multiples of \(2\pi\); the principal argument is the one in \((-\pi, \pi]\). See Chapter 9.

Argument principle A theorem of complex analysis: if no root of \(p\) lies on a circle, the number of times \(p\) winds the circle around 0 equals the number of roots inside it. See Chapter 12.

Arithmetic mean The average \(\frac{x + y}{2}\) of two numbers. For positive numbers it is never less than the geometric mean \(\sqrt{xy}\) (the AM-GM inequality). See Chapter 5.

Arithmetic sequence A sequence that adds the same number \(d\), the common difference, at each step: \(a_n = a_1 + (n - 1)d\). Its sum is the number of terms times the average of the first and last. See Chapter 8.

Associates Two elements \(a\) and \(b\) of an integral domain with \(a = ub\) for some unit \(u\), such as \(3\) and \(-3\) in \(\mathbb{Z}\). Associates divide each other and count as the same factor. See Chapter 18.

Associative law The law \((ab)c = a(bc)\) (and the same for addition): the grouping of a product does not matter. It is one of the group axioms. See Chapter 1.

Augmented matrix The table of numbers of a linear system, one row per equation, with the coefficients followed by the right-hand side. Gaussian elimination works on its rows. See Chapter 3.

Automorphism An isomorphism from a structure to itself. An automorphism of a field \(K\) is a bijective ring homomorphism \(K \to K\); an \(F\)-automorphism also fixes every element of \(F\). See Chapter 15 and Chapter 20.

Axis of symmetry The vertical line through the vertex of a parabola \(y = ax^2 + bx + c\), namely \(x = -\frac{b}{2a}\). The parabola is its own mirror image in it. See Chapter 5.

B

Base (of a power) In \(x^n\), the number \(x\) that is multiplied by itself; \(n\) is the exponent. For exponential functions \(a^x\) and logarithms \(\log_a\), the base \(a\) must be positive and not 1. See Chapter 1.

Base case The first step of a proof by induction: showing that the statement \(P(n)\) holds for the starting value of \(n\). See Chapter 8.

Basis (of a module) A list of elements of a module that generates it and is independent. A module with a basis is free, and then \(M \cong R^n\). See Chapter 21.

Bézout's theorem Two plane curves of degrees \(d\) and \(e\) with no common component meet in at most \(de\) points, and in exactly \(de\) when counted correctly over \(\mathbb{C}\) with points at infinity. See Chapter 22.

Bijection A function that is both one-to-one and onto. A bijection pairs off the elements of two sets exactly, and it is precisely a function that has an inverse. See Chapter 6.

Binary operation A rule that takes two elements \(a\) and \(b\) of a set \(G\), in order, and gives an element of \(G\). Addition and multiplication of numbers, and composition of symmetries, are examples. See Chapter 13.

Binet's formula The closed formula \(F_n = \frac{\varphi^n - \psi^n}{\sqrt5}\) for the Fibonacci numbers, where \(\varphi\) and \(\psi\) are the roots of \(r^2 = r + 1\). See Chapter 8.

Binomial A polynomial with two terms, such as \(x + 1\) or \(3x^2 - 5\). See Chapter 4.

Binomial coefficient The number \(\binom{n}{k} = \frac{n!}{k!\,(n - k)!}\), read "\(n\) choose \(k\)": the number of ways to choose \(k\) things from \(n\), and the coefficient of \(a^{n-k}b^k\) in \((a + b)^n\). See Chapter 8.

Binomial theorem The expansion \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\) for every whole number \(n\), proved by induction using Pascal's rule. See Chapter 8.

Buchberger's algorithm The algorithm that computes a Gröbner basis: form S-polynomials, reduce them by the current basis, add any nonzero remainders, and repeat until every S-polynomial reduces to 0 (Buchberger's criterion). See Chapter 22.

Burnside's lemma If a finite group \(G\) acts on a finite set \(X\), the number of orbits is the average number of points fixed by an element of \(G\). It counts objects "up to symmetry", such as necklaces and coloured cubes. See Chapter 16.

C

Cancellation law In a group, \(ab = ac\) implies \(b = c\), and \(ba = ca\) implies \(b = c\). In an integral domain the same holds for multiplication by a nonzero element. See Chapter 13.

Cardano's formula The formula that solves the depressed cubic \(t^3 + pt + q = 0\) as a sum of two cube roots, \(t = \sqrt[3]{-\frac q2 + \sqrt{D}} + \sqrt[3]{-\frac q2 - \sqrt{D}}\) with \(D = \frac{q^2}{4} + \frac{p^3}{27}\). It was published by Cardano in 1545. See Chapter 11.

Cartesian product The set \(A \times B\) of all ordered pairs \((a, b)\) with \(a \in A\) and \(b \in B\). A function from \(A\) to \(B\) can be defined as a subset of it. See Chapter 6.

Category A collection of objects and, for each pair of objects, a set of morphisms (arrows) between them, with an associative composition and identity morphisms. Sets, groups, rings and vector spaces each form a category. See Chapter 22.

Cauchy's theorem If a prime \(p\) divides the order of a finite group \(G\), then \(G\) has an element of order \(p\). See Chapter 16.

Cayley table The full multiplication table of a small finite group: the entry in row \(a\), column \(b\) is \(ab\). Each element appears exactly once in every row and every column. See Chapter 13.

Cayley's theorem Every group is isomorphic to a group of permutations: \(G\) acts faithfully on itself by left multiplication, so it sits inside the symmetric group on its own elements. See Chapter 16.

Centralizer The subgroup \(C_G(x) = \{g : gx = xg\}\) of elements that commute with \(x\). It is the stabilizer of \(x\) under conjugation. See Chapter 16.

Centre (of a group) The subgroup \(Z(G)\) of elements that commute with every element of \(G\). It is always normal, and it is all of \(G\) exactly when \(G\) is abelian. See Chapter 15.

Character The function \(\chi_\rho(g) = \operatorname{tr} \rho(g)\) that sends each group element to the trace of its matrix in a representation \(\rho\). Characters are class functions, and they determine a complex representation up to isomorphism. See Chapter 21.

Characteristic (of a ring) The smallest positive integer \(n\) with \(n \cdot 1 = 0\), or 0 if there is none. The characteristic of an integral domain is 0 or a prime. See Chapter 17.

Characteristic equation The polynomial equation obtained by trying \(a_n = r^n\) in a linear recurrence with constant coefficients; for Fibonacci it is \(r^2 = r + 1\). Its roots give the general solution. See Chapter 8.

Characteristic polynomial For a linear map \(T\), the polynomial \(\det(xI - T)\). In the module view it is the product of the invariant factors of \(V\) as an \(F[x]\)-module. See Chapter 21.

Chinese remainder theorem If \(\gcd(m, n) = 1\), each pair of congruences \(x \equiv a \pmod m\), \(x \equiv b \pmod n\) has exactly one solution mod \(mn\); equivalently \(\mathbb{Z}_{mn} \cong \mathbb{Z}_m \times \mathbb{Z}_n\). For rings, coprime ideals \(I\) and \(J\) give \(R/IJ \cong R/I \times R/J\). See Chapter 15 and Chapter 17.

Class equation For a finite group, \(|G| = |Z(G)| + \sum [G : C_G(x_i)]\), the sum running over one element from each conjugacy class of size greater than 1. It shows that every group of prime-power order has a nontrivial centre. See Chapter 16.

Class field theory The theory, built by Takagi and Artin in the 1920s, that describes how primes factor in a number field with abelian Galois group, through congruence conditions. It generalizes quadratic reciprocity. See Chapter 22.

Class function A function on a group that is constant on each conjugacy class, so \(\chi(hgh^{-1}) = \chi(g)\). Characters are class functions. See Chapter 21.

Class number The size of the ideal class group of a ring of integers. It measures how far unique factorization of elements fails, and it is 1 exactly when the ring is a PID. See Chapter 18.

Classification of finite simple groups The theorem that every finite simple group is cyclic of prime order, an alternating group \(A_n\) with \(n \ge 5\), a group of Lie type, or one of 26 sporadic groups. Its proof fills tens of thousands of pages. See Chapter 16.

Codomain A set named as the place where a function's outputs live, as in \(f : A \to B\). The range, the outputs actually produced, may be smaller. See Chapter 6.

Coefficient The number factor of a term: in \(3x^2\) it is \(3\), in \(-5xy\) it is \(-5\), and in \(x\) alone it is \(1\). See Chapter 1.

Commutative law The law \(ab = ba\) (and \(a + b = b + a\)): the order of the two numbers does not matter. Matrix multiplication and the symmetries of a square do not obey it. See Chapter 1.

Commutative ring A ring in which \(ab = ba\) for all \(a\) and \(b\), such as \(\mathbb{Z}\), \(\mathbb{Z}_n\) and \(F[x]\). Square matrices form a ring that is not commutative. See Chapter 17.

Commutator The element \(hkh^{-1}k^{-1}\) of a group; it equals \(e\) exactly when \(h\) and \(k\) commute. The subgroup the commutators generate, \([G, G]\), measures how far \(G\) is from abelian. See Chapter 15.

Companion matrix A matrix whose characteristic polynomial is a given monic polynomial, so that its eigenvalues are exactly the polynomial's roots. Numerical software finds polynomial roots this way, and companion matrices are the blocks of the rational canonical form. See Chapter 12.

Completing the square Rewriting \(x^2 + bx\) as \(\left(x + \frac b2\right)^2 - \frac{b^2}{4}\), turning a quadratic into a perfect square plus a constant. It proves the quadratic formula and gives vertex form. See Chapter 5.

Complex number An expression \(a + bi\) with \(a\) and \(b\) real and \(i^2 = -1\). The set of complex numbers, \(\mathbb{C}\), is a field that contains the real numbers. See Chapter 9.

Complex plane The plane in which the complex number \(a + bi\) is the point \((a, b)\), also called the Argand diagram. The horizontal axis holds the real numbers and the vertical axis the purely imaginary ones. See Chapter 9.

Composition (of functions) Doing one function and then feeding its output into another: \((g \circ f)(x) = g(f(x))\), "first \(f\), then \(g\)". Composition is associative but usually not commutative. See Chapter 6.

Composition series A chain of subgroups \(\{e\} = G_0 \trianglelefteq G_1 \trianglelefteq \cdots \trianglelefteq G_n = G\) in which each quotient \(G_i/G_{i-1}\) is simple. These quotients are the composition factors. See Chapter 16.

Conjugacy class The set \(\{gxg^{-1} : g \in G\}\) of all conjugates of \(x\): the orbit of \(x\) when \(G\) acts on itself by conjugation. See Chapter 16.

Conjugate (complex) The conjugate of \(z = a + bi\) is \(\bar z = a - bi\), its mirror image in the real axis. The product \(z\bar z = a^2 + b^2\) is real, which is how complex division works. See Chapter 9.

Conjugate (in a group) The element \(gkg^{-1}\) is the conjugate of \(k\) by \(g\); it is \(k\) seen from another point of view. Conjugate elements of \(S_n\) are exactly those with the same cycle shape. See Chapter 15.

Constant term A term with no letters, such as \(7\) in \(3x^2 - 5xy + 7\). In a polynomial \(a_nx^n + \cdots + a_0\) it is \(a_0\). See Chapter 1.

Constructible number A real number that is a coordinate of a point reachable from \((0, 0)\) and \((1, 0)\) with straightedge and compass. Every constructible number has degree a power of 2 over \(\mathbb{Q}\). See Chapter 19.

Content (of a polynomial) The greatest common divisor of the coefficients of a nonzero polynomial in \(\mathbb{Z}[x]\). A polynomial with content 1 is primitive. See Chapter 18.

Converge A series converges to \(S\) if its partial sums approach \(S\); a sequence converges if it has a limit. A series that does not converge diverges. See Chapter 8.

Coprime ideals Two ideals \(I\) and \(J\) of a ring with \(I + J = R\), that is, \(a + b = 1\) for some \(a \in I\) and \(b \in J\). They are the setting of the Chinese remainder theorem for rings. See Chapter 17.

Correspondence theorem The subgroups of \(G/N\) are exactly the \(H/N\) for subgroups \(H\) with \(N \le H \le G\), matched in an order-preserving way; the same holds for ideals of a quotient ring. See Chapter 15.

Coset For a subgroup \(H\) of \(G\) and \(g \in G\), the left coset \(gH = \{gh : h \in H\}\) or the right coset \(Hg\). The left cosets cut \(G\) into disjoint pieces, all the same size as \(H\). See Chapter 14.

Cramer's rule The formula for the solution of a linear system as a ratio of determinants: for two unknowns, \(x = D_x / D\) and \(y = D_y / D\) when \(D \neq 0\). See Chapter 3.

Cubic A polynomial of degree 3. Every real cubic has at least one real root, and its roots can be found by Cardano's formula. See Chapter 11.

Cycle (permutation) A permutation that moves \(a_1 \to a_2 \to \cdots \to a_k \to a_1\) and fixes everything else, written \((a_1\,a_2\,\ldots\,a_k)\): a \(k\)-cycle. Every permutation is a product of disjoint cycles. See Chapter 13.

Cyclic group A group generated by one element \(g\): every element is a power \(g^k\). Up to isomorphism the cyclic groups are \(\mathbb{Z}\) and the \(\mathbb{Z}_n\). See Chapter 13.

Cyclic subgroup The subgroup \(\langle g \rangle = \{g^k : k \in \mathbb{Z}\}\) of all powers of one element. Its order equals the order of \(g\). See Chapter 13.

Cyclotomic polynomial For a prime \(p\), the polynomial \(\Phi_p(x) = x^{p-1} + x^{p-2} + \cdots + x + 1\), whose roots are the primitive \(p\)-th roots of unity. It is irreducible over \(\mathbb{Q}\), by Eisenstein after the shift \(x \mapsto x + 1\). See Chapter 18.

D

De Moivre's theorem The formula \((\cos\theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta\): to raise a complex number to a power, raise its modulus to the power and multiply its argument by \(n\). See Chapter 9.

Dedekind domain An integral domain in which every nonzero ideal factors uniquely as a product of prime ideals. The ring of integers of every number field is one, which rescues unique factorization at the level of ideals. See Chapter 22.

Degree (of a field extension) The dimension \([K : F]\) of \(K\) as a vector space over \(F\). For example \([\mathbb{Q}(\sqrt2) : \mathbb{Q}] = 2\) and \([\mathbb{C} : \mathbb{R}] = 2\). See Chapter 19.

Degree (of a polynomial) The highest power \(n\) with a nonzero coefficient \(a_n\). The zero polynomial has no degree, and over an integral domain the degree of a product is the sum of the degrees. See Chapter 4.

Degree (of a term) The total number of letter factors in a term: \(3x^2\) and \(-5xy\) both have degree 2. See Chapter 1.

Dependent system A linear system with infinitely many solutions, because one equation adds nothing to the others; for two unknowns the two lines are the same line. See Chapter 3.

Depressed cubic A cubic with no square term, \(t^3 + pt + q\). Every cubic \(x^3 + bx^2 + cx + d\) becomes one by the substitution \(x = t - \frac b3\). See Chapter 11.

Descartes' rule of signs The number of positive roots of a real polynomial equals the number of sign changes in its coefficients, or is less than it by an even number. See Chapter 10.

Determinant (of a 2 by 2 system) The number \(D = ad - bc\) for the system \(ax + by = e\), \(cx + dy = f\). The system has exactly one solution when \(D \neq 0\). See Chapter 3.

Difference of two squares The identity \(a^2 - b^2 = (a - b)(a + b)\), one of the most used factorizations. See Chapter 4.

Dihedral group The group \(D_n\) of the \(2n\) symmetries of a regular \(n\)-gon: \(n\) rotations and \(n\) reflections. \(D_4\), the symmetries of a square, is the book's first group. See Chapter 13.

Direct product The group \(G \times H\) of pairs \((g, h)\), multiplied one coordinate at a time. The order of \((g, h)\) is the least common multiple of the orders of \(g\) and \(h\). See Chapter 15.

Directrix The fixed line \(L\) in the definition of a parabola as the set of points equally far from a point \(F\) (the focus) and from \(L\). See Chapter 5.

Discriminant (of a polynomial) For a monic polynomial with roots \(r_1, \ldots, r_n\), the product \(\prod_{i<j} (r_i - r_j)^2\). It is zero exactly when there is a repeated root, and it is a polynomial in the coefficients. See Chapter 11.

Discriminant (of a quadratic) The number \(\Delta = b^2 - 4ac\). The quadratic \(ax^2 + bx + c\) has two real roots if \(\Delta > 0\), one double root if \(\Delta = 0\), and none if \(\Delta < 0\). See Chapter 5.

Distributive law The law \(a(b + c) = ab + ac\), the one law that links addition and multiplication. It is behind expanding brackets and factoring. See Chapter 1.

Diverge A series diverges if its partial sums do not approach a number. The harmonic series diverges even though its terms go to 0. See Chapter 8.

Division algorithm (for polynomials) For polynomials \(f\) and \(g \neq 0\) over a field there are unique \(q\) and \(r\) with \(f = qg + r\) and \(r = 0\) or \(\deg r < \deg g\). See Chapter 4.

Division ring A ring in which every nonzero element has a multiplicative inverse, but multiplication need not commute. Hamilton's quaternions are the classic example. See Chapter 17.

Domain (of a function) The set of inputs a function accepts. The natural domain of a formula is the set of all inputs for which it makes sense. See Chapter 6.

Double root A root that occurs twice, as \(x = -\frac{b}{2a}\) does when the discriminant of a quadratic is 0: the factor appears squared, and the graph touches the axis there. See Chapter 5.

E

\(e\) (the number) The number \(e \approx 2.71828\), the limit of \((1 + 1/m)^m\) as \(m\) grows: the result of continuous compounding at 100% for one unit of time. It is the base of the natural logarithm. See Chapter 7.

Eisenstein's criterion If a prime \(p\) divides every coefficient of \(f \in \mathbb{Z}[x]\) except the leading one, and \(p^2\) does not divide the constant term, then \(f\) is irreducible over \(\mathbb{Q}\). See Chapter 18.

Elementary divisors The prime powers in the decomposition of a finite abelian group (or a module over a PID) into cyclic pieces of prime-power order. The list is unique up to order. See Chapter 15.

Elementary row operations The three moves of elimination: swap two equations, multiply one by a nonzero number, and add a multiple of one to another. Each gives an equivalent system. See Chapter 3.

Elementary symmetric polynomials The polynomials \(e_1 = \sum x_i\), \(e_2 = \sum_{i<j} x_ix_j\), ..., \(e_n = x_1 \cdots x_n\). By Vieta's formulas they are, up to sign, the coefficients of the monic polynomial with roots \(x_1, \ldots, x_n\). See Chapter 10.

Equation A statement that two expressions are equal, meant to be true for particular values of its unknowns. To solve it is to find its solution set. See Chapter 1.

Equivalence relation A relation that is reflexive, symmetric and transitive. Its classes form a partition; the left cosets of a subgroup are the classes of \(a \sim b \iff a^{-1}b \in H\). See Chapter 14.

Equivalent equations Equations with exactly the same solution set. The safe moves of algebra turn an equation into an equivalent one. See Chapter 2.

Euclidean domain An integral domain with a size function \(N\) that allows division with a smaller remainder: for \(a\) and \(b \ne 0\) there are \(q, r\) with \(a = qb + r\) and \(r = 0\) or \(N(r) < N(b)\). \(\mathbb{Z}\), \(F[x]\) and the Gaussian integers are examples. See Chapter 18.

Euclid's lemma If a prime \(p\) divides \(ab\), then \(p\) divides \(a\) or \(p\) divides \(b\). It is the key step in proving unique factorization. See Chapter 18.

Euler's formula The identity \(e^{i\theta} = \cos\theta + i\sin\theta\). With \(\theta = \pi\) it gives \(e^{i\pi} + 1 = 0\). See Chapter 9.

Euler's phi function The number \(\varphi(n)\) of integers from 1 to \(n\) that are coprime to \(n\); it is the order of the group \(U(n)\). Euler's theorem says \(a^{\varphi(n)} \equiv 1 \pmod n\) when \(\gcd(a, n) = 1\). See Chapter 14.

Evaluate To work out which number an expression names once the values of its letters are known. See Chapter 1.

Evaluation homomorphism The ring homomorphism \(F[x] \to K\), \(f \mapsto f(\alpha)\). Its kernel is generated by the minimal polynomial of \(\alpha\) when \(\alpha\) is algebraic. See Chapter 19.

Even and odd functions A function is even if \(f(-x) = f(x)\) (graph symmetric in the \(y\)-axis) and odd if \(f(-x) = -f(x)\) (graph symmetric under a half-turn about the origin). See Chapter 6.

Even and odd permutations A permutation is even if it is a product of an even number of transpositions and odd otherwise. The parity does not depend on how the permutation is written. See Chapter 13.

Exponent In \(x^n\), the number \(n\) that says how many copies of the base are multiplied. Exponents extend to zero, negative, fractional and finally all real values. See Chapter 7.

Exponential function A function \(f(x) = a^x\) with base \(a > 0\), \(a \neq 1\), or more generally \(f(t) = Ca^t\). Equal steps in \(t\) multiply \(f\) by equal factors. See Chapter 7.

Expression A meaningful string of numbers, letters and operation signs, such as \(3x^2 - 5xy + 7\). It names a number once its letters are given values. See Chapter 1.

Extension (of fields) If \(F \subseteq K\) are fields with the same operations, \(K\) is an extension of \(F\), written \(K/F\). \(K\) is then a vector space over \(F\). See Chapter 19.

Extraneous solution A number produced by a solving step that is not a solution of the original equation. Multiplying by an expression that could be zero, or squaring both sides, can create one. See Chapter 2.

F

Factor A piece that is multiplied: the term \(-5xy\) has factors \(-5\), \(x\) and \(y\). To factor a polynomial is to write it as a product of polynomials of lower degree. See Chapter 4.

Factor theorem A number \(c\) is a root of a polynomial \(f\) exactly when \(x - c\) is a factor of \(f\). See Chapter 10.

Faithful action An action in which only the identity moves nothing: no other element fixes every point. See Chapter 16.

Feasible region The set of points that satisfy all the constraints of a linear program. It is a convex polygon, and the best value of the objective is found at a corner. See Chapter 3.

Feit-Thompson theorem Every finite group of odd order is solvable (Feit and Thompson, 1963). Its proof took 255 pages. See Chapter 16.

Fermat's little theorem If \(p\) is prime and \(p\) does not divide \(a\), then \(a^{p-1} \equiv 1 \pmod p\). The book proves it from Lagrange's theorem. See Chapter 14.

Ferrari's method The method, found by Lodovico Ferrari, that solves the quartic by adding a well-chosen square to both sides, which leads to a cubic (the resolvent cubic). See Chapter 11.

Field A commutative ring with \(1 \neq 0\) in which every nonzero element has a multiplicative inverse. \(\mathbb{Q}\), \(\mathbb{R}\), \(\mathbb{C}\) and \(\mathbb{Z}_p\) for prime \(p\) are fields. See Chapter 1 and Chapter 17.

Finite extension A field extension \(K/F\) of finite degree \([K : F]\). Every finite extension is algebraic. See Chapter 19.

Finite field A field with finitely many elements. There is one, \(\mathbb{F}_{p^n}\), of each prime-power size and no others, and its group of nonzero elements is cyclic. See Chapter 19.

Finitely generated A module (or group) is finitely generated if some finite list of its elements generates it: every element is a combination \(r_1m_1 + \cdots + r_nm_n\). See Chapter 21.

First isomorphism theorem If \(\varphi : G \to H\) is a homomorphism, then \(G/\ker\varphi \cong \operatorname{im}\varphi\). The same holds for rings with ideals in place of normal subgroups. See Chapter 15.

Fixed field For a group \(G\) of automorphisms of a field \(K\), the subfield \(K^G\) of elements fixed by every element of \(G\). See Chapter 20.

Focus The fixed point \(F\) in the definition of a parabola as the set of points equally far from \(F\) and a line (the directrix). Rays parallel to the axis reflect through it. See Chapter 5.

Formula An equation that says how to compute one quantity from others, such as \(F = \frac95 C + 32\). See Chapter 1.

Free module A module that has a basis; it is then isomorphic to \(R^n\). Vector spaces are always free, but \(\mathbb{Z}_2\) as a \(\mathbb{Z}\)-module is not. See Chapter 21.

Frobenius map In a ring of prime characteristic \(p\), the map \(a \mapsto a^p\). It is a ring homomorphism, because \((a + b)^p = a^p + b^p\) (the "freshman's dream"), and on a finite field it is an automorphism. See Chapter 17.

Function A rule that takes an input and gives back exactly one output. Formally, a function from \(A\) to \(B\) is a subset of \(A \times B\) containing exactly one pair \((a, b)\) for each \(a \in A\). See Chapter 6.

Functor A map between categories that sends objects to objects and morphisms to morphisms, preserving composition and identities. A contravariant functor reverses the arrows. See Chapter 22.

Fundamental theorem of algebra Every nonconstant polynomial with complex coefficients has a complex root; so a polynomial of degree \(n\) splits into \(n\) linear factors over \(\mathbb{C}\). See Chapter 12.

Fundamental theorem of finite abelian groups Every finite abelian group is a direct product of cyclic groups of prime-power order, in a way that is unique up to order. See Chapter 15.

Fundamental theorem of Galois theory For a finite Galois extension \(K/F\) with group \(G\), subgroups of \(G\) and fields between \(F\) and \(K\) match one to one, reversing inclusions, with \(H \mapsto K^H\); degrees become indices, and normal subgroups match normal extensions. See Chapter 20.

Fundamental unit In the ring of integers of a real quadratic field, a unit from which all units are \(\pm\) its powers, such as \(1 + \sqrt2\) in \(\mathbb{Z}[\sqrt2]\). Finding it means solving Pell's equation. See Chapter 22.

G

Galois extension A finite extension \(K/F\) that is normal and separable; equivalently, the splitting field of a separable polynomial, or one whose automorphism group has order \([K : F]\), or one whose fixed field is exactly \(F\). See Chapter 20.

Galois group The group \(\operatorname{Gal}(K/F)\) of \(F\)-automorphisms of a Galois extension \(K/F\). The Galois group of a polynomial is that of its splitting field; it permutes the roots. See Chapter 20.

Galois representation A representation of the Galois group of a number field, usually on a vector space over a field such as the \(\ell\)-adic numbers. They are central to the proof of Fermat's last theorem and to the Langlands programme. See Chapter 21.

Gaussian elimination The method that solves a linear system by eliminating unknowns one at a time with elementary row operations, reaching triangular form, and then substituting back. See Chapter 3.

Gaussian integers The ring \(\mathbb{Z}[i] = \{a + bi : a, b \in \mathbb{Z}\}\) with norm \(N(a + bi) = a^2 + b^2\). It is a Euclidean domain, and its primes explain which numbers are sums of two squares. See Chapter 18.

Gauss's lemma The product of two primitive polynomials in \(\mathbb{Z}[x]\) is primitive. Consequently a polynomial in \(\mathbb{Z}[x]\) that factors over \(\mathbb{Q}\) already factors over \(\mathbb{Z}\). See Chapter 18.

General polynomial The polynomial \((x - t_1)(x - t_2) \cdots (x - t_n)\) with independent variables \(t_i\) as roots, over the field of symmetric rational functions in them. Its Galois group is \(S_n\), which is why it has no formula in radicals for \(n \ge 5\). See Chapter 20.

Generator An element \(g\) of a group with \(\langle g \rangle\) the whole group. A generator of the cyclic group \(U(n)\) is called a primitive root mod \(n\); a generator of the multiplicative group of a finite field is called a primitive element. See Chapter 13.

Geometric mean The number \(\sqrt{xy}\) for positive \(x\) and \(y\). It is at most the arithmetic mean, with equality exactly when \(x = y\). See Chapter 5.

Geometric sequence A sequence that multiplies by the same number \(r\), the common ratio, at each step: \(a_n = a_1r^{n-1}\). Its sum is \(a_1\frac{1 - r^n}{1 - r}\) for \(r \neq 1\). See Chapter 8.

Geometric series The infinite sum \(a + ar + ar^2 + \cdots\). It converges to \(\frac{a}{1 - r}\) when \(|r| < 1\) and diverges otherwise (for \(a \neq 0\)). See Chapter 8.

Golden ratio The number \(\varphi = \frac{1 + \sqrt5}{2} \approx 1.618\), the positive root of \(r^2 = r + 1\). Ratios of consecutive Fibonacci numbers approach it. See Chapter 8.

Graph (of a function) The set of all points \((x, f(x))\) in the plane, for \(x\) in the domain. A curve is the graph of a function of \(x\) exactly when every vertical line meets it at most once. See Chapter 6.

Gröbner basis A generating set of a polynomial ideal \(I\) such that a polynomial lies in \(I\) exactly when its remainder on division by the set is 0. Buchberger's algorithm computes one; it is the engine of computer algebra. See Chapter 22.

Group A set with a binary operation that is associative, has an identity element, and gives every element an inverse. Symmetries, numbers under addition and invertible matrices are examples. See Chapter 13.

Group algebra The ring \(\mathbb{C}[G]\) with the elements of a group \(G\) as a basis and multiplication coming from the group. Representations of \(G\) are the same as \(\mathbb{C}[G]\)-modules. See Chapter 21.

Group of Lie type A member of one of 16 infinite families of finite simple matrix groups over finite fields, such as \(PSL_n(q)\). Together with the cyclic, alternating and sporadic groups they make up the finite simple groups. See Chapter 16.

H

Half-life The fixed time \(T\) in which a decaying quantity halves; after \(t\) units a fraction \(\left(\frac12\right)^{t/T}\) is left. See Chapter 7.

Half-plane One of the two sides of a line in the plane, the solution set of a linear inequality such as \(2x + 3y \le 12\). See Chapter 3.

Harmonic series The series \(1 + \frac12 + \frac13 + \cdots\). Its terms go to 0 but it diverges, which shows that terms going to zero is not enough for convergence. See Chapter 8.

Hilbert's basis theorem If \(R\) is Noetherian, so is \(R[x]\); in particular every ideal of \(F[x_1, \ldots, x_n]\) is generated by finitely many polynomials. See Chapter 17.

Hilbert's Nullstellensatz Over an algebraically closed field \(k\): a set of polynomials has no common zero exactly when 1 is in the ideal they generate (weak form), and \(I(V(J)) = \sqrt{J}\) for every ideal \(J\) (strong form). It ties ideals to varieties. See Chapter 22.

Homomorphism (of groups) A function \(\varphi : G \to H\) with \(\varphi(ab) = \varphi(a)\varphi(b)\) for all \(a, b\): a map that respects the operation. It sends the identity to the identity and inverses to inverses. See Chapter 15.

Horner's form A polynomial rewritten as nested multiplications, such as \(((2x - 3)x + 4)x - 5\). Evaluating it this way is the same work as synthetic division. See Chapter 4.

I

Ideal A subset \(I\) of a commutative ring that is a subgroup under addition and absorbs multiplication: \(a \in I\) and \(r \in R\) give \(ra \in I\). Ideals are exactly the kernels of ring homomorphisms. See Chapter 17.

Ideal class group The group of nonzero ideals of a ring of integers modulo the principal ones. Its size, the class number, measures the failure of unique factorization of elements. See Chapter 18.

Idempotent A ring element \(a\) with \(a^2 = a\), such as 0, 1, and \(3\) and \(4\) in \(\mathbb{Z}_6\). See Chapter 17.

Identity An equation meant to hold for every value of its variables, such as \((a + b)^2 = a^2 + 2ab + b^2\). It is proved, not solved. See Chapter 1.

Identity element An element \(e\) with \(ea = ae = a\) for every \(a\): 0 for addition, 1 for multiplication, the "do nothing" symmetry for a group. In a group it is unique. See Chapter 13.

Image The set \(\operatorname{im}\varphi = \{\varphi(g) : g \in G\}\) of values of a homomorphism. It is a subgroup of the target. See Chapter 15.

Imaginary part The real number \(b\) in \(z = a + bi\), written \(\operatorname{Im} z\). Note that it is \(b\), not \(bi\). See Chapter 9.

Inconsistent system A linear system with no solution; for two unknowns the lines are parallel and distinct. See Chapter 3.

Index (of a subgroup) The number of left cosets of \(H\) in \(G\), written \([G : H]\). For finite groups \([G : H] = |G|/|H|\), and index is multiplicative in towers. See Chapter 14.

Induction A method of proof for statements \(P(n)\) about whole numbers: prove the base case, then prove that \(P(n)\) implies \(P(n + 1)\). It rests on the well-ordering principle. See Chapter 8.

Inductive hypothesis In the inductive step, the assumption that \(P(n)\) holds, from which \(P(n + 1)\) is to be shown. See Chapter 8.

Inequality A statement that one expression is less than (or greater than, or at most, or at least) another. Multiplying both sides by a negative number reverses its direction. See Chapter 2.

Integral domain A commutative ring with \(1 \ne 0\) and no zero divisors, so \(ab = 0\) forces \(a = 0\) or \(b = 0\). \(\mathbb{Z}\) and every field are integral domains; \(\mathbb{Z}_6\) is not. See Chapter 17.

Intertwiner A linear map \(\varphi\) between representations with \(\varphi\rho_V(g) = \rho_W(g)\varphi\) for all \(g\). Schur's lemma says an intertwiner between irreducible representations is 0 or an isomorphism. See Chapter 21.

Interval A whole stretch of the number line, such as \([a, b]\) (ends included), \((a, b)\) (ends excluded) or \([a, \infty)\). Solutions of inequalities are usually intervals. See Chapter 2.

Invariant factors The numbers (or ring elements) \(d_1 \mid d_2 \mid \cdots \mid d_k\) in the decomposition \(\mathbb{Z}_{d_1} \times \cdots \times \mathbb{Z}_{d_k}\) of a finite abelian group, or \(R/(d_1) \oplus \cdots\) of a module over a PID. They are uniquely determined. See Chapter 15 and Chapter 21.

Invariant subspace A subspace \(W\) with \(T(W) \subseteq W\) for a linear map \(T\), or \(\rho(g)W \subseteq W\) for every \(g\) in a representation. It is a submodule in the module view. See Chapter 21.

Inverse (in a group) For an element \(a\), the element \(a^{-1}\) with \(aa^{-1} = a^{-1}a = e\). In a group it is unique, and \((ab)^{-1} = b^{-1}a^{-1}\) (the socks and shoes rule). See Chapter 13.

Inverse function A function \(g : B \to A\) that undoes \(f : A \to B\): \(g(f(a)) = a\) and \(f(g(b)) = b\). A function has an inverse exactly when it is a bijection, and its graph is the mirror image of \(f\)'s in \(y = x\). See Chapter 6.

Irreducible element A nonzero non-unit \(p\) of an integral domain such that whenever \(p = ab\), one of \(a\) and \(b\) is a unit. In a UFD irreducibles are prime, but in \(\mathbb{Z}[\sqrt{-5}]\) the element 2 is irreducible and not prime. See Chapter 18.

Irreducible polynomial A nonconstant polynomial that is not a product of two polynomials of smaller degree over the given field. \(x^2 + 1\) is irreducible over \(\mathbb{R}\) but not over \(\mathbb{C}\). See Chapter 4 and Chapter 17.

Irreducible representation A nonzero representation whose only invariant subspaces are 0 and the whole space. By Maschke's theorem every complex representation of a finite group is a direct sum of irreducible ones. See Chapter 21.

Isomorphism A homomorphism that is a bijection. Two groups (or rings) are isomorphic, \(G \cong H\), if an isomorphism exists: they are the same structure with the elements renamed. See Chapter 15.

J

Jordan block The matrix with \(\lambda\) on the diagonal, 1 just above it, and 0 elsewhere. The Jordan normal form writes a complex linear map as a block-diagonal matrix of Jordan blocks. See Chapter 21.

Jordan-Hölder theorem The list of composition factors of a finite group, with multiplicities, does not depend on the composition series chosen. Simple groups are the atoms of finite groups. See Chapter 16.

K

Kernel The set \(\ker\varphi\) of elements a homomorphism sends to the identity (or to 0 for rings). It is a normal subgroup (or an ideal), and it is trivial exactly when \(\varphi\) is one-to-one. See Chapter 15.

Klein four-group The group \(V_4 = \{e, a, b, c\}\) in which every non-identity element has order 2 and the product of any two of them is the third. It is isomorphic to \(\mathbb{Z}_2 \times \mathbb{Z}_2\), and with \(\mathbb{Z}_4\) it is one of the two groups of order 4. See Chapter 14.

L

Lagrange interpolation The formula for the unique polynomial of degree less than \(n\) through \(n\) given points with different \(x\)-values. See Chapter 10.

Lagrange resolvent An expression such as \(r_1 + \omega r_2 + \omega^2 r_3\) in the roots of a polynomial, with \(\omega\) a root of unity. Its powers take few values under permutations of the roots, and that is why cubics and quartics can be solved. See Chapter 11.

Lagrange's theorem In a finite group the order of every subgroup divides the order of the group. It follows that the order of every element divides \(|G|\). See Chapter 14.

Leading coefficient In a polynomial of degree \(n\), the coefficient \(a_n\) of the highest power; \(a_nx^n\) is the leading term. See Chapter 4.

Lexicographic order The dictionary order on monomials: compare the powers of \(x_1\), then of \(x_2\) if they are equal, and so on. It is used to prove the theorem on symmetric polynomials and in Gröbner bases. See Chapter 10.

Lie algebra A vector space with a bilinear bracket \([X, Y]\) that is skew-symmetric and satisfies the Jacobi identity. The tangent space at the identity of a Lie group is one. See Chapter 22.

Lie group A group that is also a smooth manifold, with multiplication and inverse smooth maps, such as the rotation group \(SO(3)\). Lie groups describe continuous symmetry. See Chapter 22.

Like terms Terms with exactly the same letters raised to exactly the same powers, such as \(3x^2\) and \(-8x^2\). Only like terms can be collected into one. See Chapter 1.

Limit (of a sequence) A sequence \(s_n\) has limit \(S\) if, for every tolerance \(\varepsilon > 0\), all terms from some point on are within \(\varepsilon\) of \(S\). See Chapter 8.

Linear equation An equation in \(x\) that can be rewritten as \(ax + b = 0\). If \(a \neq 0\) it has exactly one solution, \(x = -b/a\). See Chapter 2.

Linear program A problem of making a linear objective as large or small as possible subject to linear inequality constraints. The best value is reached at a corner of the feasible region. See Chapter 3.

Linear recurrence with constant coefficients A rule such as \(a_{n+1} = pa_n + qa_{n-1}\) with fixed \(p\) and \(q\). It is solved by trying \(a_n = r^n\), which leads to the characteristic equation. See Chapter 8.

Logarithm The logarithm to base \(a\) of \(x\), \(\log_a x\), is the exponent \(y\) with \(a^y = x\): it counts how many factors of \(a\) make \(x\). It turns products into sums. See Chapter 7.

M

Maschke's theorem Every representation of a finite group over \(\mathbb{C}\) (or any field whose characteristic does not divide \(|G|\)) is a direct sum of irreducible representations: every invariant subspace has an invariant complement. See Chapter 21.

Maximal ideal An ideal \(M \ne R\) with no ideal strictly between \(M\) and \(R\). \(R/M\) is a field exactly when \(M\) is maximal. See Chapter 17.

Minimal polynomial (of an algebraic element) The monic polynomial of least degree over \(F\) with \(\alpha\) as a root. It is irreducible, it divides every polynomial with root \(\alpha\), and its degree is \([F(\alpha) : F]\). See Chapter 19.

Minimal polynomial (of a linear map) The monic polynomial of least degree with \(p(T) = 0\). It is the largest invariant factor of \(V\) as an \(F[x]\)-module. See Chapter 21.

Modular form A function on the upper half-plane with an enormous symmetry group, given by a \(q\)-series. The modularity theorem says every elliptic curve over \(\mathbb{Q}\) matches one. See Chapter 22.

Module An abelian group \(M\) with a multiplication by elements of a ring \(R\) satisfying the vector space rules. Over a field it is a vector space; over \(\mathbb{Z}\) it is an abelian group. See Chapter 21.

Modulus (of a complex number) The length \(|z| = \sqrt{a^2 + b^2}\) of \(z = a + bi\), its distance from 0. It satisfies \(|zw| = |z|\,|w|\) and \(z\bar z = |z|^2\). See Chapter 9.

Monic polynomial A polynomial whose leading coefficient is 1, such as \(x^2 - 3x + 2\). See Chapter 4.

Monomial A polynomial with one term, such as \(5x^3\), or in several variables a product such as \(x_1^{a_1} \cdots x_n^{a_n}\). See Chapter 4.

Multiplicity A root \(a\) of \(f\) has multiplicity \(m\) if \((x - a)^m\) divides \(f\) but \((x - a)^{m+1}\) does not. Counted with multiplicity, a polynomial of degree \(n\) has at most \(n\) roots. See Chapter 10.

N

Natural logarithm The logarithm to base \(e\), written \(\ln\). It is the inverse of the exponential function \(e^x\). See Chapter 7.

Natural transformation A way of mapping one functor to another that commutes with every morphism. It makes precise when an isomorphism is "natural", like a vector space and its double dual. See Chapter 22.

Newton's identities Formulas that relate the power sums \(p_k = r_1^k + \cdots + r_n^k\) of the roots to the elementary symmetric polynomials, so power sums can be found from the coefficients without the roots. See Chapter 10.

Newton's method The iteration \(x_{k+1} = x_k - \frac{f(x_k)}{f'(x_k)}\) for finding a root numerically. Near a simple root it roughly doubles the number of correct digits at each step. See Chapter 12.

Nilpotent A ring element \(a\) with \(a^n = 0\) for some \(n\), such as \(6\) in \(\mathbb{Z}_{12}\). The nilpotent elements of a commutative ring form an ideal, the nilradical. See Chapter 17.

Noetherian ring A ring in which every ideal is finitely generated; equivalently, every increasing chain of ideals stops. Named after Emmy Noether. See Chapter 17.

Norm A multiplicative size function on a ring of numbers, such as \(N(a + bi) = a^2 + b^2\) on \(\mathbb{Z}[i]\) or \(N(a + b\sqrt{-5}) = a^2 + 5b^2\). Units have norm 1, and it turns factoring questions into questions about integers. See Chapter 18.

Normal extension An algebraic extension \(K/F\) in which every irreducible polynomial over \(F\) that has one root in \(K\) splits completely in \(K\). Finite normal extensions are splitting fields. See Chapter 20.

Normal subgroup A subgroup \(N\) with \(gNg^{-1} = N\) for every \(g \in G\), written \(N \trianglelefteq G\); equivalently, its left and right cosets agree. Normal subgroups are exactly the kernels of homomorphisms, and the ones you can form quotients by. See Chapter 15.

Normalizer The subgroup \(N_G(H) = \{g : gHg^{-1} = H\}\): the stabilizer of \(H\) when \(G\) acts on its subgroups by conjugation. The number of conjugates of \(H\) is \([G : N_G(H)]\). See Chapter 16.

Number field A finite extension of \(\mathbb{Q}\), such as \(\mathbb{Q}(i)\) or \(\mathbb{Q}(\sqrt[3]2)\). Its ring of integers plays the part that \(\mathbb{Z}\) plays for \(\mathbb{Q}\). See Chapter 22.

O

Objective The quantity a linear program makes as large or as small as possible, such as profit \(P = 30x + 50y\). See Chapter 3.

One-to-one (injective) A function is one-to-one if different inputs give different outputs: \(f(a_1) = f(a_2)\) implies \(a_1 = a_2\). A homomorphism is one-to-one exactly when its kernel is trivial. See Chapter 6.

Onto (surjective) A function \(f : A \to B\) is onto if every element of \(B\) is an output: the range is the whole codomain. See Chapter 6.

Orbit The set \(G \cdot x = \{g \cdot x : g \in G\}\) of all places an action can move \(x\) to. The orbits partition the set acted on. See Chapter 16.

Orbit-stabilizer theorem For a group \(G\) acting on \(X\), the orbit of \(x\) is in one-to-one correspondence with the cosets of its stabilizer, so \(|G| = |G \cdot x| \cdot |G_x|\) for finite \(G\). See Chapter 16.

Order (of a group) The number of elements of a group, written \(|G|\): \(|D_4| = 8\), \(|S_n| = n!\), \(|U(n)| = \varphi(n)\). See Chapter 13.

Order (of an element) The smallest \(k \ge 1\) with \(g^k = e\), or infinite if there is none. It equals the order of the cyclic subgroup \(\langle g \rangle\), and \(g^m = e\) exactly when the order divides \(m\). See Chapter 13 and Chapter 14.

Order of operations The convention for reading an expression: brackets first, then powers, then multiplication and division from left to right, then addition and subtraction from left to right. See Chapter 1.

Ordered field A field with a notion of positive numbers that is closed under addition and multiplication, with every number exactly one of positive, negative or zero. \(\mathbb{Q}\) and \(\mathbb{R}\) are ordered fields; \(\mathbb{C}\) cannot be ordered. See Chapter 2.

Ordered pair A pair \((x, y)\) in which order matters, so \((2, 5) \neq (5, 2)\). Points of the plane are ordered pairs of coordinates. See Chapter 3.

Origin The point \(O = (0, 0)\) where the axes of a coordinate plane cross. See Chapter 3.

P

\(p\)-group A group whose order is a power of a prime \(p\). Every nontrivial finite \(p\)-group has a nontrivial centre. See Chapter 16.

Parabola The graph of \(y = ax^2 + bx + c\) with \(a \neq 0\); equivalently, the set of points equally far from a focus and a directrix. See Chapter 5.

Parallel lines Lines with the same slope that do not meet. Two distinct non-vertical lines are parallel exactly when their slopes are equal. See Chapter 3.

Parameter A number that is fixed in a given situation but could change from one situation to another, such as \(a\), \(b\) and \(c\) in \(ax^2 + bx + c\). See Chapter 1.

Partial fraction decomposition Writing a proper rational expression as a sum of simpler fractions with denominators that are powers of linear or irreducible quadratic factors, such as \(\frac{1}{x^2 - 1} = \frac{1/2}{x - 1} - \frac{1/2}{x + 1}\). See Chapter 4.

Partial sums For a series \(\sum a_k\), the finite sums \(s_n = a_1 + \cdots + a_n\). The series converges when its partial sums approach a limit. See Chapter 8.

Partition A splitting of a set into disjoint, nonempty pieces that cover it, such as the cosets of a subgroup. A partition of a number is a way of writing it as a sum of positive integers, which counts abelian groups of prime-power order. See Chapter 14.

Pascal's triangle The triangle of binomial coefficients: each row starts and ends with 1, and every inner number is the sum of the two above it (Pascal's rule, \(\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}\)). See Chapter 8.

Pell's equation The equation \(a^2 - db^2 = \pm 1\) in integers. Its solutions give the units of \(\mathbb{Z}[\sqrt d]\). See Chapter 22.

Periodic function A function with \(f(x + p) = f(x)\) for every \(x\) and some fixed \(p > 0\), the period. Its graph repeats every \(p\). See Chapter 6.

Permutation A one-to-one, onto function from a set to itself: a rearrangement. The permutations of \(\{1, \ldots, n\}\) form the symmetric group \(S_n\). See Chapter 13.

Perpendicular lines Lines that meet at a right angle. Two non-vertical lines are perpendicular exactly when the product of their slopes is \(-1\). See Chapter 3.

Pigeonhole principle If more than \(n\) objects go into \(n\) boxes, some box gets two; as a fact about functions, there is no one-to-one function from a larger finite set to a smaller one. See Chapter 6.

Polar form Writing a complex number as \(z = r(\cos\theta + i\sin\theta) = re^{i\theta}\), with \(r = |z|\) and \(\theta\) an argument. Multiplying complex numbers multiplies their moduli and adds their arguments. See Chapter 9.

Polynomial An expression \(a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0\) with whole-number powers of \(x\) and numbers as coefficients. Formally, an element of a polynomial ring. See Chapter 4.

Polynomial ring For a commutative ring \(R\), the ring \(R[x]\) of polynomials with coefficients in \(R\). When \(F\) is a field, \(F[x]\) has division with remainder and every ideal is principal. See Chapter 17.

Power sums The sums \(p_k = r_1^k + r_2^k + \cdots + r_n^k\) of the \(k\)-th powers of the roots. Newton's identities relate them to the coefficients. See Chapter 10.

Prime element A nonzero non-unit \(p\) of an integral domain such that \(p \mid ab\) implies \(p \mid a\) or \(p \mid b\). Every prime is irreducible. See Chapter 18.

Prime field The smallest subfield of a field: \(\mathbb{Q}\) in characteristic 0, and \(\mathbb{F}_p\) in characteristic \(p\). See Chapter 17.

Prime ideal An ideal \(P \ne R\) such that \(ab \in P\) implies \(a \in P\) or \(b \in P\). \(R/P\) is an integral domain exactly when \(P\) is prime, and every maximal ideal is prime. See Chapter 17.

Primitive element A generator of the cyclic group of nonzero elements of a finite field. Separately, an element \(\alpha\) with \(K = F(\alpha)\); the primitive element theorem says one exists for every finite extension in characteristic 0. See Chapter 19.

Primitive polynomial A polynomial in \(\mathbb{Z}[x]\) whose coefficients have greatest common divisor 1. See Chapter 18.

Primitive root (mod \(n\)) A generator of the group \(U(n)\) of units mod \(n\). One exists exactly when \(n\) is \(1\), \(2\), \(4\), \(p^k\) or \(2p^k\) for an odd prime \(p\). See Chapter 13.

Primitive root of unity An \(n\)-th root of unity \(\omega\) whose powers give all \(n\) of them, such as \(e^{2\pi i/n}\). See Chapter 9.

Principal ideal The ideal \((a) = \{ra : r \in R\}\) of all multiples of one element. In \(\mathbb{Z}\) and in \(F[x]\) every ideal is principal. See Chapter 17.

Principal ideal domain (PID) An integral domain in which every ideal is principal. Every Euclidean domain is a PID, and every PID is a UFD; in a PID gcds exist and satisfy Bézout's identity. See Chapter 18.

Proper rational expression A fraction \(\frac{p}{q}\) of polynomials with \(\deg p < \deg q\). Partial fractions apply to proper fractions; an improper one is divided first. See Chapter 4.

Proper subgroup A subgroup other than the trivial subgroup \(\{e\}\) and the whole group. See Chapter 14.

Purely imaginary A complex number \(bi\) with real part 0. See Chapter 9.

Q

Quadrant One of the four regions into which the axes cut the coordinate plane, numbered counterclockwise from the top right. See Chapter 3.

Quadratic A polynomial of degree 2, \(ax^2 + bx + c\) with \(a \neq 0\). Its graph is a parabola. See Chapter 5.

Quadratic formula The roots of \(ax^2 + bx + c = 0\) are \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), proved by completing the square. See Chapter 5.

Quartic A polynomial of degree 4. Ferrari's method reduces it to a cubic, so quartics can be solved by radicals. See Chapter 11.

Quaternion group The group \(Q_8 = \{\pm 1, \pm i, \pm j, \pm k\}\) with \(i^2 = j^2 = k^2 = ijk = -1\). It is a nonabelian group of order 8 in which every subgroup is normal. See Chapter 14.

Quaternions Hamilton's numbers \(a + bi + cj + dk\) with \(i^2 = j^2 = k^2 = ijk = -1\), found in 1843. They form a division ring that is not commutative; unit quaternions describe rotations in space. See Chapter 9 and Chapter 17.

Quintic A polynomial of degree 5. The general quintic has no formula in radicals, and particular ones such as \(x^5 - 4x + 2\) cannot be solved by radicals. See Chapter 20.

Quotient group For a normal subgroup \(N\) of \(G\), the group \(G/N\) of cosets of \(N\), with \((aN)(bN) = abN\). Also called a factor group; \(\mathbb{Z}/n\mathbb{Z} = \mathbb{Z}_n\) is the basic example. See Chapter 15.

Quotient ring For an ideal \(I\) of \(R\), the ring \(R/I\) of cosets \(a + I\), with the operations of \(R\). \(\mathbb{Z}/(n) = \mathbb{Z}_n\) and \(\mathbb{R}[x]/(x^2 + 1) \cong \mathbb{C}\) are examples. See Chapter 17.

R

Radical (of an ideal) The ideal \(\sqrt{J}\) of polynomials (or ring elements) \(f\) with \(f^r \in J\) for some \(r \ge 1\). The strong Nullstellensatz says \(I(V(J)) = \sqrt{J}\). See Chapter 22.

Radical extension An extension \(L/F\) reached by a chain of fields \(F = L_0 \subseteq L_1 \subseteq \cdots \subseteq L_m = L\), each obtained from the one before by adjoining an \(n\)-th root of one of its elements. See Chapter 20.

Range The set of outputs a function actually produces. It lies inside the codomain and may be smaller. See Chapter 6.

Rank (of a module) The number \(r\) of copies of \(R\) in the decomposition \(M \cong R^r \oplus R/(d_1) \oplus \cdots \oplus R/(d_k)\) of a finitely generated module over a PID: the size of its free part. See Chapter 21.

Rate of change How fast \(y\) changes as \(x\) changes; for a line it is the slope, with units of \(y\) per unit of \(x\). See Chapter 3.

Rational canonical form A block-diagonal matrix of companion matrices, one for each invariant factor of a linear map. It exists over any field, with no need to find eigenvalues. See Chapter 21.

Rational expression A fraction \(\frac{p(x)}{q(x)}\) of two polynomials, with \(q\) not the zero polynomial. You may cancel common factors, never common terms. See Chapter 4.

Rational root theorem If a polynomial with integer coefficients has a rational root \(\frac pq\) in lowest terms, then \(p\) divides the constant term and \(q\) divides the leading coefficient. See Chapter 10.

Real part The real number \(a\) in \(z = a + bi\), written \(\operatorname{Re} z\). See Chapter 9.

Reciprocal For a number \(a \neq 0\), the number \(\frac1a\) with \(a \cdot \frac1a = 1\): its inverse under multiplication. Zero has no reciprocal. See Chapter 1.

Recurrence A rule that builds each term of a sequence from earlier ones, such as \(F_{n+1} = F_n + F_{n-1}\) for the Fibonacci numbers. See Chapter 8.

Regular representation The representation of a group \(G\) on the group algebra \(\mathbb{C}[G]\) by left multiplication, of dimension \(|G|\). It contains every irreducible representation, as many times as its dimension. See Chapter 21.

Relation A set of ordered pairs in which an element may be paired with several others, or none, such as "is less than". A function is a special kind of relation. See Chapter 6.

Remainder theorem The remainder when a polynomial \(f(x)\) is divided by \(x - c\) is \(f(c)\). See Chapter 10.

Representation A homomorphism \(\rho : G \to GL(V)\) from a group to the invertible linear maps of a vector space, so each group element acts as a matrix with \(\rho(gh) = \rho(g)\rho(h)\). Its dimension is \(\dim V\). See Chapter 21.

Resolvent cubic The cubic equation that Ferrari's method needs to solve on the way to the roots of a quartic. See Chapter 11.

Ring A set with addition and multiplication such that it is an abelian group under addition, multiplication is associative with an identity 1, and the distributive laws hold. \(\mathbb{Z}\), \(\mathbb{Z}_n\), polynomials and square matrices are rings. See Chapter 17.

Ring homomorphism A map \(\varphi : R \to S\) with \(\varphi(a + b) = \varphi(a) + \varphi(b)\), \(\varphi(ab) = \varphi(a)\varphi(b)\) and \(\varphi(1) = 1\). Its kernel is an ideal. See Chapter 17.

Ring of integers The ring \(\mathcal{O}_K\) of algebraic integers in a number field \(K\), such as \(\mathbb{Z}[i]\) in \(\mathbb{Q}(i)\). It is a Dedekind domain, but it need not have unique factorization of elements. See Chapter 22.

Root A number that makes an equation true, or at which a polynomial is 0. Also called a solution or, for a polynomial, a zero. See Chapter 2.

Roots of unity The \(n\) complex numbers \(z\) with \(z^n = 1\), namely \(e^{2\pi ik/n}\) for \(k = 0, \ldots, n - 1\). They sit at the corners of a regular \(n\)-gon on the unit circle. See Chapter 9.

S

S-polynomial For two polynomials, the combination that multiplies each by the monomial needed to make their leading terms equal and subtracts, cancelling those terms. Buchberger's algorithm is built on it. See Chapter 22.

Semidirect product A twisted version of the direct product, \(N \rtimes H\), formed when only one of the two subgroups is normal. \(S_3 \cong \mathbb{Z}_3 \rtimes \mathbb{Z}_2\), and the nonabelian group of order 21 is \(\mathbb{Z}_7 \rtimes \mathbb{Z}_3\). See Chapter 15.

Separable A polynomial is separable if it has no repeated roots in a splitting field; an algebraic extension is separable if every element's minimal polynomial is. In characteristic 0 everything is separable. See Chapter 20.

Sequence A list of numbers in order, \(a_1, a_2, a_3, \ldots\), given by an explicit formula or by a recurrence. The numbers are its terms. See Chapter 8.

Series An infinite sum \(\sum_{k=1}^{\infty} a_k\), defined as the limit of its partial sums when that limit exists. See Chapter 8.

Sigma notation The notation \(\sum_{k=1}^{n} a_k\) for \(a_1 + a_2 + \cdots + a_n\). The letter \(k\) is a dummy index with no meaning outside the sum. See Chapter 8.

Simple extension A field extension \(F(\alpha)\) generated by one element. If \(\alpha\) is algebraic, \(F(\alpha) \cong F[x]/(m(x))\) with \(m\) its minimal polynomial. See Chapter 19.

Simple group A nontrivial group whose only normal subgroups are \(\{e\}\) and itself. The cyclic groups of prime order and \(A_n\) for \(n \ge 5\) are simple. See Chapter 16.

Slope For the line through \((x_1, y_1)\) and \((x_2, y_2)\) with \(x_1 \neq x_2\), the number \(m = \frac{y_2 - y_1}{x_2 - x_1}\): rise over run. Vertical lines have no slope. See Chapter 3.

Smith normal form The diagonal form \(\operatorname{diag}(d_1, d_2, \ldots)\) with \(d_1 \mid d_2 \mid \cdots\) that row and column operations reach for any matrix over a Euclidean domain. It proves the structure theorem for modules over a PID. See Chapter 21.

Solution set The collection of all solutions of an equation or inequality. To solve is to find it. See Chapter 2.

Solvable by radicals A polynomial \(f\) over \(F\) is solvable by radicals if its splitting field lies in some radical extension of \(F\): its roots can be written with \(+\), \(-\), \(\times\), \(\div\) and \(n\)-th roots. In characteristic 0 this happens exactly when its Galois group is solvable. See Chapter 20.

Solvable group A group with a chain of subgroups \(\{e\} = G_0 \trianglelefteq G_1 \trianglelefteq \cdots \trianglelefteq G_n = G\) in which every quotient \(G_{i}/G_{i-1}\) is abelian. \(S_4\) is solvable; \(S_5\) is not. See Chapter 20.

Splitting field An extension of \(F\) in which a polynomial \(f\) factors into linear factors, and which is generated over \(F\) by the roots of \(f\). It exists and is unique up to isomorphism. See Chapter 19.

Sporadic groups The 26 finite simple groups that fit into none of the infinite families. The largest is the Monster, of order about \(8 \times 10^{53}\). See Chapter 16.

Stabilizer The subgroup \(G_x = \{g : g \cdot x = x\}\) of elements that leave \(x\) where it is. See Chapter 16.

Strong induction A form of induction in which the inductive step may assume \(P(k)\) for every \(k \le n\), not only \(P(n)\), to prove \(P(n + 1)\). See Chapter 8.

Structure theorem for modules over a PID Every finitely generated module over a PID is \(R^r \oplus R/(d_1) \oplus \cdots \oplus R/(d_k)\) with \(d_1 \mid \cdots \mid d_k\), uniquely. It gives the classification of finitely generated abelian groups and the Jordan normal form. See Chapter 21.

Subgroup A subset \(H\) of a group that is itself a group with the same operation: it contains the identity and is closed under products and inverses. Written \(H \le G\). See Chapter 14.

Subgroup generated by a set For \(S \subseteq G\), the smallest subgroup \(\langle S \rangle\) containing \(S\): all products of elements of \(S\) and their inverses. See Chapter 14.

Submodule A subgroup of a module that is closed under multiplication by ring elements. See Chapter 21.

Subring A subset of a ring that contains 1 and is closed under subtraction and multiplication, so it is a ring with the same operations. See Chapter 17.

Substitute To replace each letter of an expression by a given number, putting every substituted number in brackets. See Chapter 1.

Sylow subgroup For \(|G| = p^km\) with \(p \nmid m\), a subgroup of order \(p^k\), the largest power of \(p\) possible. Sylow's theorems say such subgroups exist, are all conjugate, and number \(n_p \equiv 1 \pmod p\) with \(n_p \mid m\). See Chapter 16.

Symmetric group The group \(S_n\) of all permutations of \(\{1, 2, \ldots, n\}\) under composition, with \(n!\) elements. Every finite group is isomorphic to a subgroup of some \(S_n\). See Chapter 13.

Symmetric polynomial A polynomial in several variables that stays the same when the variables are permuted. Every symmetric polynomial is a polynomial in the elementary symmetric polynomials. See Chapter 10.

Synthetic division A shortcut for dividing a polynomial by \(x - c\) that works only with the coefficients. Also called Ruffini's rule, it is the same computation as evaluating in Horner's form. See Chapter 4.

System of linear equations A collection of linear equations to be satisfied at the same time. It has exactly one solution, none (inconsistent) or infinitely many (dependent). See Chapter 3.

T

Term A piece of an expression that is added or subtracted: \(3x^2 - 5xy + 7\) has the three terms \(3x^2\), \(-5xy\) and \(7\). The numbers in a sequence are also called its terms. See Chapter 1.

Torsion An element \(m\) of a module is torsion if \(rm = 0\) for some nonzero \(r\) in the ring. A module is torsion-free if 0 is its only torsion element, and torsion if every element is. See Chapter 21.

Tower law For fields \(F \subseteq K \subseteq L\), \([L : F] = [L : K]\,[K : F]\): degrees multiply. See Chapter 19.

Transcendental number A number that is not algebraic over \(\mathbb{Q}\): it is a root of no nonzero polynomial with rational coefficients. Liouville's number, \(e\) and \(\pi\) are transcendental, and almost every real number is. See Chapter 19.

Transitive action An action with only one orbit: any point can be moved to any other. See Chapter 16.

Transposition A 2-cycle \((a\,b)\), which swaps two elements and fixes the rest. Every permutation is a product of transpositions. See Chapter 13.

Triangle inequality The inequality \(|a + b| \le |a| + |b|\), for real numbers and for complex numbers alike: one side of a triangle is never longer than the other two together. See Chapter 2 and Chapter 9.

Triangular number The number \(1 + 2 + \cdots + n = \frac{n(n + 1)}{2}\), so called because that many dots form a triangle. See Chapter 8.

Trinomial A polynomial with three terms, such as \(x^2 + 5x + 6\). See Chapter 4.

Trivial subgroup The subgroup \(\{e\}\) containing only the identity. Every group has it and itself as subgroups. See Chapter 14.

U

Unique factorization domain (UFD) An integral domain in which every nonzero non-unit is a product of irreducibles, uniquely up to order and associates. \(\mathbb{Z}\), \(F[x]\) and every PID are UFDs; \(\mathbb{Z}[\sqrt{-5}]\) is not, since \(6 = 2 \cdot 3 = (1 + \sqrt{-5})(1 - \sqrt{-5})\). See Chapter 18.

Unit An element of a ring with a multiplicative inverse. The units of \(\mathbb{Z}\) are \(\pm 1\), those of \(\mathbb{Z}[i]\) are \(\pm 1, \pm i\), and in a field every nonzero element is a unit. See Chapter 17.

Units mod \(n\) The group \(U(n)\) of residues mod \(n\) that are coprime to \(n\), under multiplication. It has \(\varphi(n)\) elements. See Chapter 13.

Universal property A description of an object by how it maps to or from every other object, which determines it up to unique isomorphism. Direct products and quotients have universal properties. See Chapter 22.

Unknown A letter that stands for one particular number (or a few) that we are trying to find, such as \(x\) in \(3x + 2 = 17\). See Chapter 1.

V

Variable A letter that stands for any number from some collection, in a statement meant to hold for each of them, such as \(a\) and \(b\) in \(a + b = b + a\). See Chapter 1.

Vertex The turning point of a parabola, its highest or lowest point. For \(y = a(x - h)^2 + k\) it is \((h, k)\). See Chapter 5.

Vertex form The form \(y = a(x - h)^2 + k\) of a quadratic, found by completing the square. It shows the vertex \((h, k)\) and so the maximum or minimum value. See Chapter 5.

Vertical line test A curve in the plane is the graph of a function of \(x\) exactly when every vertical line meets it at most once. See Chapter 6.

Vieta's formulas For a monic polynomial with roots \(r_1, \ldots, r_n\), the coefficients are, up to sign, the elementary symmetric polynomials of the roots; for a quadratic, the roots add to \(-\frac ba\) and multiply to \(\frac ca\). See Chapter 10.

W

Well-ordering principle Every nonempty set of whole numbers has a smallest element. It is the reason proof by induction is valid. See Chapter 8.

Wilson's theorem For a prime \(p\), \((p - 1)! \equiv -1 \pmod p\): in \(\mathbb{F}_p\) each nonzero element pairs with its inverse, leaving only \(1\) and \(-1\). See Chapter 18.

Winding number For a closed path in the plane that misses 0, the number of times it goes around 0 counterclockwise. It drives the winding-number proof of the fundamental theorem of algebra. See Chapter 12.

X

\(x\)-intercept A point where a graph crosses the \(x\)-axis, found by putting \(y = 0\). For the graph of a polynomial these are its real roots. See Chapter 3.

Y

\(y\)-intercept The point where a graph crosses the \(y\)-axis, found by putting \(x = 0\); for \(y = mx + b\) it is \((0, b)\). See Chapter 3.

Z

Zero divisor A nonzero element \(a\) of a commutative ring with \(ab = 0\) for some nonzero \(b\), such as \(2\) and \(3\) in \(\mathbb{Z}_6\). Integral domains have none. See Chapter 17.

Zero polynomial The polynomial \(0\), with no nonzero coefficients. It has no degree (or degree \(-\infty\) by convention). See Chapter 4.

Zero product property If \(ab = 0\), then \(a = 0\) or \(b = 0\). It is why factoring solves equations, and it is exactly what an integral domain requires. See Chapter 1.

\(\mathbb{Z}_n\) (integers mod \(n\)) The numbers \(0, 1, \ldots, n - 1\) with addition and multiplication mod \(n\). Under addition it is the cyclic group of order \(n\); as a ring it is a field exactly when \(n\) is prime. See Chapter 13 and Chapter 17.