In abstract algebra, a normal extension is an algebraic field extension L/K for which every irreducible polynomial over K that has a root in L splits into linear factors in L.^[1][2] These are one of the conditions for algebraic extensions to be a Galois extension. Bourbaki calls such an extension a quasi-Galois extension.

Definition

Let ${\displaystyle L/K}$ be an algebraic extension (i.e., L is an algebraic extension of K), such that ${\displaystyle L\subseteq {\overline {K))}$ (i.e., L is contained in an algebraic closure of K). Then the following conditions, any of which can be regarded as a definition of normal extension, are equivalent:^[3]

• Every embedding of L in ${\displaystyle {\overline {K))}$ over K induces an automorphism of L.
• L is the splitting field of a family of polynomials in ${\displaystyle K[X]}$.
• Every irreducible polynomial of ${\displaystyle K[X]}$ that has a root in L splits into linear factors in L.

Other properties

Let L be an extension of a field K. Then:

• If L is a normal extension of K and if E is an intermediate extension (that is, L ⊇ E ⊇ K), then L is a normal extension of E.^[4]
• If E and F are normal extensions of K contained in L, then the compositum EF and E ∩ F are also normal extensions of K.^[4]

Equivalent conditions for normality

Let ${\displaystyle L/K}$ be algebraic. The field L is a normal extension if and only if any of the equivalent conditions below hold.

• The minimal polynomial over K of every element in L splits in L;
• There is a set ${\displaystyle S\subseteq K[x]}$ of polynomials that each splits over L, such that if ${\displaystyle K\subseteq F\subsetneq L}$ are fields, then S has a polynomial that does not split in F;
• All homomorphisms ${\displaystyle L\to {\bar {K))}$ that fix all elements of K have the same image;
• The group of automorphisms, ${\displaystyle {\text{Aut))(L/K),}$ of L that fix all elements of K, acts transitively on the set of homomorphisms ${\displaystyle L\to {\bar {K))}$ that fix all elements of K.

Examples and counterexamples

For example, ${\displaystyle \mathbb {Q} ({\sqrt {2)))}$ is a normal extension of ${\displaystyle \mathbb {Q} ,}$ since it is a splitting field of ${\displaystyle x^{2}-2.}$ On the other hand, ${\displaystyle \mathbb {Q} ({\sqrt[{3}]{2)))}$ is not a normal extension of ${\displaystyle \mathbb {Q} }$ since the irreducible polynomial ${\displaystyle x^{3}-2}$ has one root in it (namely, ${\displaystyle {\sqrt[{3}]{2))}$), but not all of them (it does not have the non-real cubic roots of 2). Recall that the field ${\displaystyle {\overline {\mathbb {Q} ))}$ of algebraic numbers is the algebraic closure of ${\displaystyle \mathbb {Q} ,}$ and thus it contains ${\displaystyle \mathbb {Q} ({\sqrt[{3}]{2))).}$ Let ${\displaystyle \omega }$ be a primitive cubic root of unity. Then since,

$${\displaystyle \mathbb {Q} ({\sqrt[{3}]{2)))=\left.\left\{a+b{\sqrt[{3}]{2))+c{\sqrt[{3}]{4))\in {\overline {\mathbb {Q} ))\,\,\right|\,\,a,b,c\in \mathbb {Q} \right\))$$

$${\displaystyle {\begin{cases}\sigma :\mathbb {Q} ({\sqrt[{3}]{2)))\longrightarrow {\overline {\mathbb {Q} ))\\a+b{\sqrt[{3}]{2))+c{\sqrt[{3}]{4))\longmapsto a+b\omega {\sqrt[{3}]{2))+c\omega ^{2}{\sqrt[{3}]{4))\end{cases))}$$

${\displaystyle \mathbb {Q} ({\sqrt[{3}]{2)))}$

${\displaystyle {\overline {\mathbb {Q} ))}$

${\displaystyle \mathbb {Q} }$

${\displaystyle \sigma }$

${\displaystyle \mathbb {Q} ({\sqrt[{3}]{2))).}$

For any prime ${\displaystyle p,}$ the extension ${\displaystyle \mathbb {Q} ({\sqrt[{p}]{2)),\zeta _{p})}$ is normal of degree ${\displaystyle p(p-1).}$ It is a splitting field of ${\displaystyle x^{p}-2.}$ Here ${\displaystyle \zeta _{p))$ denotes any ${\displaystyle p}$th primitive root of unity. The field ${\displaystyle \mathbb {Q} ({\sqrt[{3}]{2)),\zeta _{3})}$ is the normal closure (see below) of ${\displaystyle \mathbb {Q} ({\sqrt[{3}]{2))).}$

Normal closure

If K is a field and L is an algebraic extension of K, then there is some algebraic extension M of L such that M is a normal extension of K. Furthermore, up to isomorphism there is only one such extension that is minimal, that is, the only subfield of M that contains L and that is a normal extension of K is M itself. This extension is called the normal closure of the extension L of K.

If L is a finite extension of K, then its normal closure is also a finite extension.