Automorphisms and Galois Theory

(Section X) Automorphisms and Galois Theory

(Section 48) Automorphisms of Fields.

(subsection) The conjugation isomorphisms of Algebraic Field Theory

(Def) all algebraic extensions and all elements algebraic over a field F under consideration are contained in one fixed algebraic closure \bar{F} of F.

(Def 48.1) Let E be an algebraic extension of a field F. Two elements \alpha , \beta \in E are conjugate over F if irr( \alpha , F ) = irr ( \beta , F ) , that is, if \alpha and \beta are zeros of the same irreducible polynomial over F.

(The Conjugation Isomorphisms) (Thm 48.3) Let F be a field, and let \alpha and \beta be algebraic over F with deg ( \alpha , ) = n. The map \psi_{\alpha , \beta } : F( \alpha ) \to F( \beta ) defined by \psi_{\alpha , \beta } (c_{0} + c_{i} \alpha + \cdots + c_{n-1} \alpha^{n-1} ) = c_{0} + c_{1} \beta + \cdots + c_{n-1} \beta^{n-1} for c_{i} \in F is an isomorphism of F( \alpha ) onto F( \beta ) iff \alpha and \beta are conjugate over F.

(Cor 48.5) Let \alpha be algebraic over a field F. Every isomorphism \psi mapping F( \alpha ) onto a subfield of \bar{F} s.t. \psi (a) = a for a \in F maps \alpha onto a conjugate \beta of \alpha over F. Conversely, for each conjugate \beta of \alpha over F, there exists exactly one isomorphism \psi_{\alpha , \beta } of F( \alpha ) onto a subfield of \bar{F} mapping \alpha onto \beta and mapping each a \in F onto itself.

(Cor 48.6) Let f(x) \in \mathbb{R} [x] . If f(a + bi) = 0 for (a + bi) \in \mathbb{C}, where a, b \in \mathbb{R}, then f(a – bi) = 0 also. Loosely, complex zeros of polynomials with real coefficient occur in conjugate pairs.

(subsection) Automorphisms and Fixed Fields

(Def 48.8) An isomorphism of a field into itself is an automorphism of the field.

(Def 48.9) If \sigma is an isomorphism of a field E onto some field, then an element a of E is left fixed by \sigma if \sigma (a) = a. A collection S of isomorphisms of E leaves a subfield F of E fixed if each a \in F is left fixed by every \sigma \in S. If { \sigma } leaves F fixed, then \sigma leaves F fixed.

(Thm 48.11) Let \( \lbrace \sigma_{i} \vert i \in I \rbrace \) be a collection of automorphisms of a field E. Then the set \(E_{\lbrace \sigma_{i} \rbrace }\) of all a \in E left fixed by every \sigma_{i} for i \in I forms a subfield of E.

(Def 48.12) The field \(E_{\lbrace \sigma_{i} \rbrace }\) of (Thm 48.11) is the fixed field of \( \lbrace \sigma_{i} \vert i \in I \rbrace \) . For a single automorphism \sigma, we shall refer to E_{ {\sigma } } as the fixed field of \sigma.

(Thm 48.14) The set of all automorphisms of a field E is a group under function composition.

(Thm 48.15) Let E be a field, and let F be a subfield of E, Then the set G( E/F) of all automorphisms of E leaving F fixed forms a subgroup of the group of all automorphisms of E. Furthermore, F \le E_{ G(E/F) } .

(Def 48.16) The group G(E/F) of the preceding theorem is the group of automorphisms of E leaving F fixed, or more briefly, the group of E over F.

(subsection) The Frobenius Automorphism

(Thm 48.19) Let F be a finite field of characteristic p. Then the map \sigma_{p} : F \to F defined by \sigma_{p} (a) = a^{p} for a \in F is an automorphism, the Frobenius automorphism, of F. Also, F_{ \sigma_{p} } \simeq \mathbb{Z}_{p} .