7.2 Galois extensions and their properties

Galois extensions are a key concept in field theory, combining normality and separability. They're crucial for understanding the relationship between field extensions and their automorphism groups, forming the foundation of Galois theory.

The Fundamental Theorem of Galois Theory establishes a one-to-one correspondence between intermediate fields of a Galois extension and subgroups of its Galois group. This powerful result connects algebra and number theory, revealing deep insights into field structures.

Galois Extensions: Definition and Properties

Definition and Examples

• A Galois extension is a field extension $L/K$ such that $L$ is the splitting field of a separable polynomial $f(x) \in K[x]$
• Examples of Galois extensions:
  • $\mathbb{Q}(\sqrt{2})/\mathbb{Q}$
  • $\mathbb{Q}(i)/\mathbb{Q}$
  • The splitting field of $x^n - 1$ over $\mathbb{Q}$ for any positive integer $n$
• Examples of non-Galois extensions:
  • $\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q}$ is not normal
  • $\mathbb{F}_p(t^p)/\mathbb{F}_p(t)$ is not separable, where $\mathbb{F}_p$ is the field with $p$ elements and $t$ is transcendental over $\mathbb{F}_p$

Equivalent Characterizations

• For a field extension $L/K$, the following conditions are equivalent to $L/K$ being Galois:
  • $L/K$ is a normal extension (every irreducible polynomial in $K[x]$ that has a root in $L$ splits completely in $L[x]$) and a separable extension (every element of $L$ is separable over $K$)
  • $L$ is the splitting field of a separable polynomial $f(x) \in K[x]$
  • $L$ is the fixed field of $\text{Aut}(L/K)$, the group of automorphisms of $L$ that fix $K$

Equivalence of Galois Extension Characterizations

Fundamental Theorem of Galois Theory

• The Fundamental Theorem of Galois Theory establishes the equivalence of the three characterizations of Galois extensions:
  1. $L/K$ is a normal and separable extension
  2. $L$ is the splitting field of a separable polynomial $f(x) \in K[x]$
  3. $L$ is the fixed field of $\text{Aut}(L/K)$

Proving the Equivalence

• To prove $(1) \Rightarrow (2)$:
  • Construct a separable polynomial $f(x) \in K[x]$ by taking the product of the minimal polynomials of a basis for $L$ over $K$
  • Show that $L$ is the splitting field of $f(x)$
• To prove $(2) \Rightarrow (3)$:
  • Use the fact that automorphisms permute roots of polynomials to show that $L$ is fixed by $\text{Aut}(L/K)$
• To prove $(3) \Rightarrow (1)$:
  • Use the properties of the fixed field and the fact that the elements of $L$ are the roots of polynomials in $K[x]$ to show that $L/K$ is normal and separable

Properties of Galois Extensions: Normality vs Separability

Normality

• A normal extension $L/K$ is a field extension such that every irreducible polynomial in $K[x]$ that has a root in $L$ splits completely in $L[x]$
• Examples of normal extensions:
  • $\mathbb{Q}(\sqrt{2}, i)/\mathbb{Q}$
  • The splitting field of any polynomial over a field

Separability

• A separable extension $L/K$ is a field extension such that every element of $L$ is separable over $K$ (i.e., its minimal polynomial over $K$ has distinct roots)
• In characteristic $0$, every extension is separable
• In characteristic $p > 0$, an extension $L/K$ is separable if and only if $L$ is linearly disjoint from $K^{(1/p)}$ over $K$
• Examples of separable extensions:
  • $\mathbb{Q}(\sqrt{2})/\mathbb{Q}$
  • $\mathbb{F}_p(t)/\mathbb{F}_p$, where $t$ is transcendental over $\mathbb{F}_p$

Properties of Galois Extensions

• The composition of Galois extensions is Galois. If $L/K$ and $M/L$ are Galois, then $M/K$ is Galois
• The Galois group $\text{Aut}(L/K)$ of a Galois extension $L/K$ is a finite group whose order is equal to the degree $[L:K]$ of the extension

Galois Correspondence: Subgroups vs Intermediate Fields

Fundamental Theorem of Galois Theory

• The Fundamental Theorem of Galois Theory establishes a one-to-one correspondence between the intermediate fields of a Galois extension $L/K$ and the subgroups of the Galois group $\text{Aut}(L/K)$

Correspondence between Subgroups and Intermediate Fields

• For each intermediate field $M$ ($K \subseteq M \subseteq L$), the corresponding subgroup is $\text{Aut}(L/M)$, the group of automorphisms of $L$ that fix $M$
• For each subgroup $H \subseteq \text{Aut}(L/K)$, the corresponding intermediate field is the fixed field $L^H = \{x \in L : \sigma(x) = x \text{ for all } \sigma \in H\}$

Properties of the Correspondence

• The correspondence reverses inclusions:
  • If $M_1 \subseteq M_2$ are intermediate fields, then $\text{Aut}(L/M_2) \subseteq \text{Aut}(L/M_1)$ as subgroups of $\text{Aut}(L/K)$
• The degree $[L:M]$ of an intermediate field $M$ is equal to the index $[\text{Aut}(L/K):\text{Aut}(L/M)]$ of its corresponding subgroup in the Galois group
• Normal subgroups of $\text{Aut}(L/K)$ correspond to Galois extensions $M/K$ within the extension $L/K$