Normal extension

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,

the map is an embedding of ${\displaystyle \mathbb {Q} ({\sqrt[{3}]{2}})}$ in ${\displaystyle {\overline {\mathbb {Q} }}}$ whose restriction to ${\displaystyle \mathbb {Q} }$ is the identity. However, ${\displaystyle \sigma }$ is not an automorphism of ${\displaystyle \mathbb {Q} ({\sqrt[{3}]{2}}).}$

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.

See also

• Galois extension
• Normal basis

Citations

References

• Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556
• Jacobson, Nathan (1989), Basic Algebra II (2nd ed.), W. H. Freeman, ISBN 0-7167-1933-9, MR 1009787