5.2 Fixed fields and the Fundamental Theorem of Galois Theory

Fixed fields and the Fundamental Theorem of Galois Theory are key concepts in understanding field extensions. They link automorphisms, subfields, and subgroups, providing a powerful framework for analyzing field structures.

This correspondence allows us to translate problems between group theory and field theory. By studying subgroups of the Galois group, we can uncover properties of subfields and vice versa, simplifying complex algebraic questions.

Fixed fields of automorphisms

Definition and properties

• The fixed field of a group of automorphisms $G$ of a field extension $K/F$ consists of all elements of $K$ fixed by every automorphism in $G$
• If $G$ is a group of automorphisms of a field $K$, the fixed field of $G$, denoted by $Fix(G)$ or $K^G$, forms a subfield of $K$
• The fixed field equals the intersection of all fields fixed by each individual automorphism in $G$
• The degree of the extension $K/Fix(G)$ matches the order of the group $G$, expressed as $[K:Fix(G)] = |G|$
• The Galois group of the extension $K/Fix(G)$ is isomorphic to the group $G$

Characterization and examples

• For a cyclic group $G = \langle \sigma \rangle$ generated by an automorphism $\sigma$, the fixed field $Fix(G)$ consists of elements $a \in K$ satisfying $\sigma(a) = a$
• In the extension $\mathbb{Q}(\sqrt{2}, i)/\mathbb{Q}$, the fixed field of the automorphism $\sigma: \sqrt{2} \mapsto -\sqrt{2}, i \mapsto i$ is $\mathbb{Q}(i)$
• For the extension $\mathbb{Q}(\sqrt[4]{2})/\mathbb{Q}$, the fixed field of the Galois group $G = \{\text{id}, \sigma, \sigma^2, \sigma^3\}$, where $\sigma: \sqrt[4]{2} \mapsto i\sqrt[4]{2}$, is $\mathbb{Q}$
• In the extension $\mathbb{Q}(\sqrt{2}, \sqrt{3})/\mathbb{Q}$, the fixed field of the subgroup $H = \{\text{id}, \sigma\}$, where $\sigma: \sqrt{2} \mapsto -\sqrt{2}, \sqrt{3} \mapsto \sqrt{3}$, is $\mathbb{Q}(\sqrt{3})$

Fundamental Theorem of Galois Theory

Statement and interpretation

• The Fundamental Theorem of Galois Theory establishes a one-to-one correspondence between subfields of a Galois extension and subgroups of its Galois group
• For a Galois extension $K/F$ with Galois group $G$, there exists a bijection between subfields of $K$ containing $F$ and subgroups of $G$, given by $H \mapsto Fix(H)$ and $E \mapsto Gal(K/E)$
• The correspondence reverses inclusions: for subgroups $H_1$ and $H_2$ of $G$ with $H_1 \subseteq H_2$, we have $Fix(H_2) \subseteq Fix(H_1)$
• The degree of the extension $K/E$ equals the index of the corresponding subgroup $Gal(K/E)$ in $G$, expressed as $[K:E] = [G:Gal(K/E)]$
• A subfield $E$ is a Galois extension of $F$ if and only if the corresponding subgroup $Gal(K/E)$ is a normal subgroup of $G$, in which case the Galois group of $E/F$ is isomorphic to the quotient group $G/Gal(K/E)$

Examples and consequences

• In the extension $\mathbb{Q}(\sqrt{2}, i)/\mathbb{Q}$, the subgroup $H = \{\text{id}, \sigma\}$, where $\sigma: \sqrt{2} \mapsto -\sqrt{2}, i \mapsto i$, corresponds to the subfield $\mathbb{Q}(i)$
• For the extension $\mathbb{Q}(\sqrt[4]{2})/\mathbb{Q}$ with Galois group $G = \{\text{id}, \sigma, \sigma^2, \sigma^3\}$, the subgroup $H = \{\text{id}, \sigma^2\}$ corresponds to the subfield $\mathbb{Q}(\sqrt{2})$
• The Fundamental Theorem implies that for a Galois extension $K/F$, the subfield $F$ corresponds to the entire Galois group $G$, while the field $K$ corresponds to the trivial subgroup $\{\text{id}\}$
• As a consequence of the Fundamental Theorem, a finite extension $K/F$ is Galois if and only if $|Aut(K/F)| = [K:F]$

Galois correspondence

Finding corresponding subfields and subgroups

• To find the subfield corresponding to a given subgroup $H$ of the Galois group $G$, calculate the fixed field $Fix(H)$ by identifying elements of $K$ fixed by every automorphism in $H$
• To find the subgroup corresponding to a given subfield $E$ of $K$ containing $F$, calculate the Galois group $Gal(K/E)$ by finding automorphisms of $K$ that fix every element of $E$
• The Galois correspondence preserves the lattice structure of subgroups and subfields: the intersection of subgroups corresponds to the compositum of subfields, and the join of subgroups corresponds to the intersection of subfields
• The Galois correspondence helps determine the lattice of subfields of a Galois extension by studying the lattice of subgroups of its Galois group

Examples and applications

• In the extension $\mathbb{Q}(\sqrt{2}, \sqrt{3})/\mathbb{Q}$, the subgroup $H = \{\text{id}, \sigma\}$, where $\sigma: \sqrt{2} \mapsto -\sqrt{2}, \sqrt{3} \mapsto \sqrt{3}$, corresponds to the subfield $\mathbb{Q}(\sqrt{3})$
• For the extension $\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q}$ with Galois group $G = \{\text{id}, \sigma, \sigma^2\}$, where $\sigma: \sqrt[3]{2} \mapsto \omega\sqrt[3]{2}$ and $\omega$ is a primitive cube root of unity, the subfield $\mathbb{Q}(\omega)$ corresponds to the subgroup $\{\text{id}, \sigma\}$
• The Galois correspondence can be used to prove that for a Galois extension $K/F$ with Galois group $G$, the number of subfields of $K$ containing $F$ equals the number of subgroups of $G$
• In cryptography, the Galois correspondence is applied to construct finite fields with desired properties for use in encryption and error correction codes

Applying the Fundamental Theorem

Solving problems in field theory

• Use the Fundamental Theorem to determine the existence and uniqueness of subfields with specific properties, such as degree or Galois group
• Apply the Galois correspondence to construct field extensions with desired properties by finding appropriate subgroups of the Galois group
• Utilize the Fundamental Theorem to prove theorems about the structure of field extensions, such as the existence of intermediate fields or the solvability of polynomial equations
• Employ the Galois correspondence to simplify the computation of Galois groups by working with subfields instead of automorphisms
• Use the Fundamental Theorem to study relationships between different field extensions and their Galois groups, such as compositums, intersections, and quotients

Examples and applications

• Prove that for a Galois extension $K/F$ with Galois group $G$, there exists a unique subfield $E$ of $K$ containing $F$ with $[E:F] = n$ if and only if $G$ has a unique subgroup of index $n$
• Construct a field extension $K/\mathbb{Q}$ with Galois group isomorphic to the dihedral group $D_4$ by finding a polynomial whose splitting field has $D_4$ as its Galois group and applying the Galois correspondence
• Use the Fundamental Theorem to prove that a polynomial $f(x) \in F[x]$ is solvable by radicals if and only if its Galois group over $F$ is a solvable group
• Determine the Galois group of the splitting field of $x^4 - 2$ over $\mathbb{Q}$ by studying the lattice of subfields and applying the Galois correspondence
• Investigate the relationship between the splitting fields of polynomials $f(x)$ and $g(x)$ over a field $F$ by examining the subgroups of the Galois group of the compositum of their splitting fields