8.2 Correspondence between subfields and subgroups

The Galois correspondence is a powerful tool that connects subfields and subgroups in field extensions. It establishes a one-to-one relationship between intermediate fields of a Galois extension and subgroups of its Galois group, preserving inclusion and degree.

This correspondence forms the heart of the Fundamental Theorem of Galois Theory. It allows us to study field extensions by examining group structures, and vice versa, providing deep insights into the algebraic relationships between fields and their automorphisms.

Galois Correspondence

Establishing the One-to-One Correspondence

• The Fundamental Theorem of Galois Theory establishes a one-to-one correspondence between the intermediate fields of a Galois extension and the subgroups of its Galois group
• For a Galois extension $L/K$ with Galois group $G$, there exists a bijective map $φ: \{intermediate fields of L/K\} → \{subgroups of G\}$ given by $φ(E) = Gal(L/E)$ for each intermediate field $E$
  • The map $φ$ associates each intermediate field $E$ with its corresponding subgroup $Gal(L/E)$ of the Galois group $G$
• The inverse map $φ^{(-1)}: \{subgroups of G\} → \{intermediate fields of L/K\}$ is given by $φ^{(-1)}(H) = L^H$ (the fixed field of $H$) for each subgroup $H$ of $G$
  • The inverse map $φ^{(-1)}$ associates each subgroup $H$ of the Galois group $G$ with its corresponding fixed field $L^H$

Preservation of Inclusion and Degree

• The correspondence preserves inclusion: if $E₁ ⊆ E₂$ are intermediate fields, then $φ(E₂) ⊆ φ(E₁)$ as subgroups of $G$
  • If one intermediate field is contained in another, their corresponding subgroups have the reverse inclusion relation
• Similarly, if $H₁ ⊆ H₂$ are subgroups of $G$, then $φ^{(-1)}(H₁) ⊇ φ^{(-1)}(H₂)$ as intermediate fields
  • If one subgroup is contained in another, their corresponding intermediate fields have the reverse inclusion relation
• The degree of an intermediate field $E$ over $K$ is equal to the index of its corresponding subgroup $φ(E)$ in $G$: $[E:K] = [G:φ(E)]$
  • This relation connects the degree of an intermediate field with the index of its corresponding subgroup in the Galois group

Intermediate Fields and Subgroups

Determining the Galois Correspondence

• To find the Galois correspondence, first determine if the given field extension $L/K$ is Galois by checking if it is both normal and separable
  • A field extension is normal if every irreducible polynomial in $K[x]$ that has a root in $L$ splits completely in $L[x]$
  • A field extension is separable if every element of $L$ is separable over $K$, meaning its minimal polynomial over $K$ has distinct roots
• Compute the Galois group $G = Gal(L/K)$ by finding all the $K$-automorphisms of $L$
  • A $K$-automorphism of $L$ is a field automorphism of $L$ that fixes every element of $K$
• Identify all the intermediate fields $E$ of $L/K$ and all the subgroups $H$ of $G$
  • Intermediate fields are fields that lie between $K$ and $L$ in the field extension $L/K$
  • Subgroups are non-empty subsets of $G$ that are closed under the group operation and taking inverses

Finding Corresponding Subgroups and Intermediate Fields

• For each intermediate field $E$, find its corresponding subgroup $φ(E) = Gal(L/E)$ by determining the $K$-automorphisms of $L$ that fix $E$
  • The subgroup $Gal(L/E)$ consists of all $K$-automorphisms of $L$ that map elements of $E$ to themselves
• For each subgroup $H$ of $G$, find its corresponding intermediate field $φ^{(-1)}(H) = L^H$ by computing the fixed field of $H$
  • The fixed field $L^H$ is the set of all elements in $L$ that are fixed by every automorphism in $H$
• Verify that the correspondence preserves inclusion and that $[E:K] = [G:φ(E)]$ for each pair of corresponding intermediate fields and subgroups
  • This step ensures that the Galois correspondence is indeed a one-to-one correspondence with the desired properties

Lattice Structure of Galois Correspondence

Isomorphic Lattices of Intermediate Fields and Subgroups

• The Galois correspondence preserves the lattice structure of intermediate fields and subgroups, forming two isomorphic lattices
  • A lattice is a partially ordered set in which every pair of elements has a unique least upper bound (join) and a unique greatest lower bound (meet)
• The lattice of intermediate fields is ordered by inclusion (⊆), with the join operation being the compositum of fields (∨) and the meet operation being the intersection of fields (∧)
  • The compositum $E₁ ∨ E₂$ is the smallest field containing both $E₁$ and $E₂$
  • The intersection $E₁ ∧ E₂$ is the largest field contained in both $E₁$ and $E₂$
• The lattice of subgroups is ordered by inclusion (⊆), with the join operation being the generated subgroup (〈〉) and the meet operation being the intersection of subgroups (∩)
  • The generated subgroup $〈H₁, H₂〉$ is the smallest subgroup containing both $H₁$ and $H₂$
  • The intersection $H₁ ∩ H₂$ is the largest subgroup contained in both $H₁$ and $H₂$

Reversal of Order and Extremal Elements

• The Galois correspondence reverses the order of the lattices: if $E₁ ⊆ E₂$ are intermediate fields, then $φ(E₂) ⊆ φ(E₁)$ as subgroups, and if $H₁ ⊆ H₂$ are subgroups, then $φ^{(-1)}(H₁) ⊇ φ^{(-1)}(H₂)$ as intermediate fields
  • The inclusion relation between intermediate fields and subgroups is reversed under the Galois correspondence
• The minimal element of the intermediate fields lattice is $K$, corresponding to the maximal element $G$ of the subgroups lattice
  • The base field $K$ is the smallest intermediate field, and its corresponding subgroup is the entire Galois group $G$
• The maximal element of the intermediate fields lattice is $L$, corresponding to the minimal element $\{1\}$ of the subgroups lattice
  • The Galois extension $L$ is the largest intermediate field, and its corresponding subgroup is the trivial subgroup $\{1\}$ consisting only of the identity element

Applications of Galois Correspondence

Determining Number and Degree of Intermediate Fields

• Use the Galois correspondence to determine the number and degree of intermediate fields for a given Galois extension based on the subgroups of its Galois group
  • The number of intermediate fields is equal to the number of subgroups of the Galois group
  • The degree of each intermediate field over the base field is equal to the index of its corresponding subgroup in the Galois group

Proving the Fundamental Theorem of Algebra

• Utilize the Galois correspondence to prove the Fundamental Theorem of Algebra by showing that the splitting field of a separable polynomial is a Galois extension
  • The Fundamental Theorem of Algebra states that every non-constant polynomial with complex coefficients has a complex root
  • The splitting field of a polynomial is the smallest field extension in which the polynomial splits into linear factors
  • A separable polynomial is a polynomial whose roots are distinct

Constructing Field Extensions with Prescribed Galois Groups

• Apply the Galois correspondence to construct examples of field extensions with prescribed Galois groups, such as cyclic or symmetric groups
  • A cyclic group is a group generated by a single element
  • The symmetric group $S_n$ is the group of all permutations of $n$ elements

Determining Solvability by Radicals

• Employ the Galois correspondence to determine the solvability of polynomial equations by radicals based on the solvability of their Galois groups
  • A polynomial equation is solvable by radicals if its roots can be expressed using arithmetic operations and taking roots (square roots, cube roots, etc.)
  • A group is solvable if it has a composition series with abelian factors

Studying the Absolute Galois Group

• Use the Galois correspondence to study the structure of the absolute Galois group Gal(K̅/K) for a given field $K$, where K̅ is its algebraic closure
  • The algebraic closure of a field $K$ is the smallest algebraically closed field containing $K$
  • The absolute Galois group of $K$ is the Galois group of its algebraic closure over $K$, which encodes important information about the field $K$