Normal and Galois Closure of a Field Extension

Subjects: Field Theory
Links: Splitting Fields and Normal Field Extensions, Algebraic Closure of a Field

Def: Let $L/F$ be an algebraic field extension. A normal closure of $L/F$ is a field extension $N$ of $L$, $F\subseteq L\subseteq N$, such that

• $N/F$ is normal.
• If $F\subseteq L\subseteq M\subseteq N$, and $M/F$ is normal, then $M=N$.
  In other words, $N$ is the smallest extension of $L$ such that $N/F$ is normal.

Lemma: If $L/F$ is finite field extension, then:

• There is a normal closure $N$ of $L/F$, and even more, $N/F$ is finite.
• If $N^{\prime }$ is a another normal closure $L/F$, the the extensions $N/F$ and $N^{\prime }/F$ are isomorphic as field extensions.

Lemma: Let $K/F$ be a finite field extension, and fix an algebraic closure $\bar{F}$ of $F$. Let ${\text{Hom}}_F(K,\bar{F})$ be the set of all $F$-embeddings of $K$ into $\bar{F}$. The composite field of $\{\sigma (K)\mid \sigma \in {\text{Hom}}_F(K,\bar{F})\}$ is the normal closure of $K/F$.

Def: If $L/F$ is an algebraic and separable field extension, then the normal closure $N$ of $L/F$, then $N/F$ is a Galois field extension, and $N$ is called the Galois closure of $L/F$.

We can prove that this Galois extension exists using Composite Field Extensions.

Prop: Let $K/F$ be a finite extension. Then $K=F(\theta )$ iff there exists only finitely many subfields of $K$ containing $F$.

The Primitive Element Theorem: If $K/F$ is finite and separable, then $K/F$ is simple. In particular, any finite extension of fields of characteristic $0$ is simple.

Prop: Let $f(x)\in F[x]$ be an irreducible polynomials of degree $b$ over the field $F$, let $L$ be the splitting field of $f(x)$ over $F$ and let $\alpha$ be a root of $f(x)$ in $L$. If $K$ is any Galois extension of $F$, then the polynomial $f(x)$ splits into a product of $m$ irreducible polynomials each of degree $d$ over $K$, where $d=[K(\alpha ):K]=[(L\cap K)(\alpha ):L\cap K]$ and $m=n/d=[F(\alpha )\cap K:F]$.

Prop: Let $L/K$ and $K/F$ are $p$-extensions, then the Galois closure of $L/F$ is also a $p$-extension.