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 over L.^[1]^[2] This is one of the conditions for an algebraic extension to be a Galois extension. Bourbaki calls such an extension a quasi-Galois extension. For finite extensions, a normal extension is identical to a splitting field.

Definition

Let $L / K$ be an algebraic extension (i.e., L is an algebraic extension of K), such that $L \subseteq \overline{K}$ (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 $\overline{K}$ over K induces an automorphism of L.
L is the splitting field of a family of polynomials in $K [X]$ .
Every irreducible polynomial of $K [X]$ 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 $L / K$ 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 $S \subseteq K [x]$ of polynomials that each splits over L, such that if $K \subseteq F ⊊ L$ are fields, then S has a polynomial that does not split in F;
All homomorphisms $L \to \bar{K}$ that fix all elements of K have the same image;
The group of automorphisms, $Aut (L / K),$ of L that fix all elements of K, acts transitively on the set of homomorphisms $L \to \bar{K}$ that fix all elements of K.

Examples and counterexamples

For example, $ℚ (\sqrt{2})$ is a normal extension of $ℚ,$ since it is a splitting field of $x^{2} - 2.$ On the other hand, $ℚ (\sqrt[3]{2})$ is not a normal extension of $ℚ$ since the irreducible polynomial $x^{3} - 2$ has one root in it (namely, $\sqrt[3]{2}$ ), but not all of them (it does not have the non-real cubic roots of 2). Recall that the field $\overline{ℚ}$ of algebraic numbers is the algebraic closure of $ℚ,$ and thus it contains $ℚ (\sqrt[3]{2}) .$ Let $ω$ be a primitive cubic root of unity. Then since, $ℚ (\sqrt[3]{2}) = {a + b \sqrt[3]{2} + c \sqrt[3]{4} \in \overline{ℚ} | a, b, c \in ℚ}$ the map ${\begin{matrix} σ : ℚ (\sqrt[3]{2}) ⟶ \overline{ℚ} \\ a + b \sqrt[3]{2} + c \sqrt[3]{4} ⟼ a + b ω \sqrt[3]{2} + c ω^{2} \sqrt[3]{4} \end{matrix}$ is an embedding of $ℚ (\sqrt[3]{2})$ in $\overline{ℚ}$ whose restriction to $ℚ$ is the identity. However, $σ$ is not an automorphism of $ℚ (\sqrt[3]{2}) .$

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

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.

Citations

↑ Lang 2002, p. 237, Theorem 3.3, NOR 3.
↑ Jacobson 1989, p. 489, Section 8.7.
↑ Lang 2002, p. 237, Theorem 3.3.
1 2 Lang 2002, p. 238, Theorem 3.4.

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

[FOOTNOTELang2002237Theorem_3.3,_NOR_3-1] Lang 2002, p. 237, Theorem 3.3, NOR 3.

[FOOTNOTEJacobson1989489Section_8.7-2] Jacobson 1989, p. 489, Section 8.7.

[FOOTNOTELang2002237Theorem_3.3-3] Lang 2002, p. 237, Theorem 3.3.

[FOOTNOTELang2002238Theorem_3.4-4] 1 2 Lang 2002, p. 238, Theorem 3.4.

[1]

[2]

[3]

[4]