Field automorphisms are the backbone of Galois Theory. They're special functions that shuffle elements of a field while keeping its structure intact. Think of them as secret codes that rearrange numbers but still let you do math with them.

These automorphisms form groups, which are like clubs of functions that play well together. By studying these groups, we can unlock hidden relationships between fields and their subfields. It's like having a master key to understand how different number systems fit together.

Field automorphisms

Definition and properties

• A field automorphism is a bijective homomorphism from a field to itself that preserves the field operations of addition and multiplication
  • Bijective means the function is both one-to-one (injective) and onto (surjective)
  • Homomorphism means the function preserves the algebraic structure of the field
• The identity map on a field, which maps every element to itself, is always an automorphism
• The composition of two field automorphisms, obtained by applying one automorphism followed by the other, is again a field automorphism
• The inverse of a field automorphism, which "undoes" the automorphism, is also a field automorphism
• Field automorphisms fix the prime subfield elementwise
  • The prime subfield is the smallest subfield of a field (e.g., $\mathbb{Q}$ for $\mathbb{R}$ or $\mathbb{C}$)
  • Elementwise means each element of the prime subfield is mapped to itself by the automorphism

Examples of field automorphisms

• The complex conjugation map $z \mapsto \overline{z}$ is an automorphism of the complex numbers $\mathbb{C}$
  • It fixes the real numbers $\mathbb{R}$ elementwise
• The Frobenius automorphism $x \mapsto x^p$ is an automorphism of a finite field $\mathbb{F}_{p^n}$ of characteristic $p$
  • It fixes the prime subfield $\mathbb{F}_p$ elementwise
• For a Galois extension $L/K$, any $K$-automorphism of $L$ (i.e., an automorphism of $L$ that fixes $K$ elementwise) is a field automorphism of $L$

Automorphism groups of field extensions

Automorphism group of a field

• The set of all automorphisms of a field forms a group under function composition, called the automorphism group of the field
  • Function composition is associative, and the identity map serves as the identity element
  • The inverse of an automorphism is also an automorphism, ensuring closure
• The automorphism group of a field extension $L/K$ is the subgroup of the automorphism group of $L$ consisting of automorphisms that fix $K$ elementwise
  • These automorphisms are also called $K$-automorphisms of $L$
• The automorphism group of a finite field $\mathbb{F}_{p^n}$ is a cyclic group generated by the Frobenius automorphism
  • The order of this group is $n$, the degree of the extension $\mathbb{F}_{p^n}/\mathbb{F}_p$

Automorphism group and splitting fields

• The automorphism group of a splitting field of a separable polynomial $f(x)$ over $K$ is isomorphic to a subgroup of the permutation group of the roots of $f(x)$
  • A splitting field is the smallest field extension of $K$ in which $f(x)$ factors into linear factors
  • The permutation group of the roots consists of all permutations of the roots that preserve the algebraic relationships among them
• For a Galois extension $L/K$, the automorphism group of $L/K$ is called the Galois group of the extension
  • The Galois group acts faithfully on the roots of the minimal polynomial of any primitive element of the extension

Applications of field automorphisms

Conjugacy and orbits

• Field automorphisms can be used to prove the transitivity of conjugacy for field extensions
  • If $\alpha$ and $\beta$ are conjugate over $K$, and $\beta$ and $\gamma$ are conjugate over $K$, then $\alpha$ and $\gamma$ are also conjugate over $K$
• Automorphisms can be applied to equations and polynomials to obtain new solutions or polynomials with the same splitting field
  • If $\sigma$ is an automorphism and $\alpha$ is a solution to a polynomial equation, then $\sigma(\alpha)$ is also a solution
• The orbit of an element under the action of the automorphism group can provide insights into the structure of the field extension
  • The orbit of $\alpha$ is the set ${\sigma(\alpha) : \sigma \in \text{Aut}(L/K)}$, where $\text{Aut}(L/K)$ is the automorphism group of $L/K$
  • The orbits partition the field extension into disjoint sets

Fixed fields and subfields

• The fixed field of a subgroup $H$ of the automorphism group of a field extension $L/K$ is the subfield of $L$ consisting of elements fixed by every automorphism in $H$
  • $\text{Fix}(H) = {x \in L : \sigma(x) = x \text{ for all } \sigma \in H}$
• For a Galois extension $L/K$, the Fundamental Theorem of Galois Theory establishes a one-to-one correspondence between the intermediate fields of $L/K$ and the subgroups of the Galois group of $L/K$
  • The correspondence associates each intermediate field $M$ with the subgroup $\text{Aut}(L/M)$ and each subgroup $H$ with the fixed field $\text{Fix}(H)$

Theorems for field automorphisms

Artin's Lemma and the Theorem of the Primitive Element

• Artin's Lemma states that if a field automorphism $\sigma$ fixes a subfield $K$ and an element $\alpha$ is algebraic over $K$, then $\sigma$ permutes the conjugates of $\alpha$ over $K$
  • The conjugates of $\alpha$ are the roots of the minimal polynomial of $\alpha$ over $K$
• The Theorem of the Primitive Element asserts that a finite separable extension $L/K$ is simple if and only if there exists a primitive element
  • A simple extension is an extension generated by a single element, i.e., $L = K(\alpha)$ for some $\alpha \in L$
  • A primitive element is an element that generates the entire extension

Dedekind's Lemma and the Fundamental Theorem of Galois Theory

• Dedekind's Lemma proves that if $M/L$ and $L/K$ are Galois extensions, then $M/K$ is Galois if and only if the automorphism groups of $M/L$ and $L/K$ intersect trivially
  • The intersection of the automorphism groups is $\text{Aut}(M/L) \cap \text{Aut}(L/K) = {\text{id}_M}$, where $\text{id}_M$ is the identity automorphism on $M$
• The Fundamental Theorem of Galois Theory establishes a correspondence between intermediate fields of a Galois extension and subgroups of its Galois group
  • The correspondence preserves inclusions and degrees: if $H_1 \subseteq H_2$ are subgroups of the Galois group, then $\text{Fix}(H_2) \subseteq \text{Fix}(H_1)$ and $[L:\text{Fix}(H_1)] = |H_1|$