Appendix E Notation
The following table defines the notation used in this book. Page numbers or references refer to the first appearance of each symbol.
| Symbol | Description | Location |
|---|---|---|
| \(\mathbb H\) | the ring of quaternions | Example 1.1.7 |
| \(\mathbb Z[i]\) | the Gaussian integers | Example 1.2.1 |
| \(\chr R\) | characteristic of a ring \(R\) | Paragraph |
| \(\mathbb Z_{(p)}\) | ring of integers localized at \(p\) | Exercise 1.6.33 |
| \(\deg f(x)\) | degree of a polynomial | Paragraph |
| \(R[x]\) | ring of polynomials over a ring \(R\) | Paragraph |
| \(R[x_1, x_2, \ldots, x_n]\) | ring of polynomials in \(n\) indeterminants | Paragraph |
| \(\phi_\alpha\) | evaluation homomorphism at \(\alpha\) | Theorem 2.1.5 |
| \(\mathbb Q(x)\) | field of rational functions over \(\mathbb Q\) | Example 3.1.5 |
| \(\nu(a)\) | Euclidean valuation of \(a\) | Paragraph |
| \(F(x)\) | field of rational functions in \(x\) | Item 3.3.7.a |
| \(F(x_1, \dots, x_n)\) | field of rational functions in \(x_1, \ldots, x_n\) | Item 3.3.7.b |
| \(\dim V\) | dimension of a vector space \(V\) | Paragraph |
| \(U \oplus V\) | direct sum of vector spaces \(U\) and \(V\) | Item 4.4.17.b |
| \(\Hom(V, W)\) | set of all linear transformations from \(U\) into \(V\) | Item 4.4.18.a |
| \(V^*\) | dual of a vector space \(V\) | Item 4.4.18.b |
| \(F( \alpha_1, \ldots, \alpha_n)\) | smallest field containing \(F\) and \(\alpha_1, \ldots, \alpha_n\) | Paragraph |
| \([E:F]\) | dimension of a field extension of \(E\) over \(F\) | Paragraph |
| \(\gf(p^n)\) | Galois field of order \(p^n\) | Paragraph |
| \(S_n\) | the symmetric group on \(n\) letters | Paragraph |
| \(D_n\) | the dihedral group | Paragraph |
| \([G:H]\) | index of a subgroup \(H\) in a group \(G\) | Paragraph |
| \(\langle a \rangle\) | cyclic subgroup generated by \(a\) | Paragraph |
| \(|a|\) | the order of an element \(a\) | Paragraph |
| \(F^*\) | multiplicative group of a field \(F\) | Paragraph |
| \(\ker \phi\) | kernel of \(\phi\) | Paragraph |
| \(G/N\) | quotient group of \(G\) by \(N\) | Paragraph |
| \({\mathcal O}_x\) | orbit of \(x\) | Paragraph |
| \(X_g\) | fixed point set of \(g\) | Paragraph |
| \(G_x\) | isotropy subgroup of \(x\) | Paragraph |
| \(G(E/F)\) | Galois group of \(E\) over \(F\) | Paragraph |
| \(F_{\{\sigma_i \}}\) | field fixed by the automorphism \(\sigma_i\) | Proposition 8.2.1 |
| \(F_G\) | field fixed by the automorphism group \(G\) | Corollary 8.2.2 |
| \(\Delta^2\) | discriminant of a polynomial | Exercise 8.3.22 |
| \((a_1, a_2, \ldots, a_k )\) | cycle of length \(k\) | Paragraph |
| \(A_n\) | the alternating group on \(n\) letters | Paragraph |
| \(N(H)\) | normalizer of s subgroup \(H\) | Paragraph |
| \(a \in A\) | \(a\) is in the set \(A\) | Paragraph |
| \({\mathbb N}\) | the natural numbers | Paragraph |
| \({\mathbb Z}\) | the integers | Paragraph |
| \({\mathbb Q}\) | the rational numbers | Paragraph |
| \({\mathbb R}\) | the real numbers | Paragraph |
| \({\mathbb C}\) | the complex numbers | Paragraph |
| \(A \subset B\) | \(A\) is a subset of \(B\) | Paragraph |
| \(\emptyset\) | the empty set | Paragraph |
| \(A \cup B\) | the union of sets \(A\) and \(B\) | Paragraph |
| \(A \cap B\) | the intersection of sets \(A\) and \(B\) | Paragraph |
| \(A'\) | complement of the set \(A\) | Paragraph |
| \(A \setminus B\) | difference between sets \(A\) and \(B\) | Paragraph |
| \(A \times B\) | Cartesian product of sets \(A\) and \(B\) | Paragraph |
| \(A^n\) | \(A \times \cdots \times A\) (\(n\) times) | Paragraph |
| \(id\) | identity mapping | Paragraph |
| \(f^{-1}\) | inverse of the function \(f\) | Paragraph |
| \(a \equiv b \pmod{n}\) | \(a\) is congruent to \(b\) modulo \(n\) | Example A.2.30 |
| \(n!\) | \(n\) factorial | Example B.1.4 |
| \(\binom{n}{k}\) | binomial coefficient \(n!/(k!(n-k)!)\) | Example B.1.4 |
| \(a \mid b\) | \(a\) divides \(b\) | Paragraph |
| \(\gcd(a, b)\) | greatest common divisor of \(a\) and \(b\) | Paragraph |
| \(\mathcal P(X)\) | power set of \(X\) | Exercise B.3.12 |
| \(\lcm(m,n)\) | the least common multiple of \(m\) and \(n\) | Exercise B.3.23 |
