Normal extension

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

Examples and counterexamples

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

Q ( 2 3 ) = { a + b 2 3 + c 4 3 ∈ Q ¯ | a , b , c ∈ Q } {\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\}}
the map
{ σ : Q ( 2 3 ) ⟶ Q ¯ a + b 2 3 + c 4 3 ⟼ a + b ω 2 3 + c ω 2 4 3 {\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}}}
is an embedding of Q ( 2 3 ) {\displaystyle \mathbb {Q} ({\sqrt[{3}]{2}})} in Q ¯ {\displaystyle {\overline {\mathbb {Q} }}} whose restriction to Q {\displaystyle \mathbb {Q} } is the identity. However, σ {\displaystyle \sigma } is not an automorphism of Q ( 2 3 ) . {\displaystyle \mathbb {Q} ({\sqrt[{3}]{2}}).}

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

  1. ^ Lang 2002, p. 237, Theorem 3.3, NOR 3.
  2. ^ Jacobson 1989, p. 489, Section 8.7.
  3. ^ Lang 2002, p. 237, Theorem 3.3.
  4. ^ a b 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