Normal extension

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In abstract algebra, an algebraic field extension L/K is said to be normal if L is the splitting field of a family of polynomials in K[X]. Bourbaki calls such an extension a quasi-Galois extension.

Equivalent properties and examples

The normality of L/K is equivalent to either of the following properties. Let Ka be an algebraic closure of K containing L.

  • Every embedding σ of L in Ka that restricts to the identity on K, satisfies σ(L) = L. In other words, σ is an automorphism of L over K.
  • Every irreducible polynomial in K[X] that has one root in L, has all of its roots in L, that is, it decomposes into linear factors in L[X]. (One says that the polynomial splits in L.)

If L is a finite extension of K that is separable (for example, this is automatically satisfied if K is finite or has characteristic zero) then the following property is also equivalent:

  • There exists an irreducible polynomial whose roots, together with the elements of K, generate L. (One says that L is the splitting field for the polynomial.)

For example, {\mathbb {Q}}({\sqrt {2}}) is a normal extension of {\mathbb {Q}}, since it is a splitting field of x2 − 2. On the other hand, {\mathbb {Q}}({\sqrt[ {3}]{2}}) is not a normal extension of {\mathbb {Q}} since the irreducible polynomial x3 − 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).

The fact that {\mathbb {Q}}({\sqrt[ {3}]{2}}) is not a normal extension of {\mathbb {Q}} can also be seen using the first of the three properties above. The field {\mathbb {A}} of algebraic numbers is an algebraic closure of {\mathbb {Q}} containing {\mathbb {Q}}({\sqrt[ {3}]{2}}). On the other hand

{\mathbb {Q}}({\sqrt[ {3}]{2}})=\{a+b{\sqrt[ {3}]{2}}+c{\sqrt[ {3}]{4}}\in {\mathbb {A}}\,|\,a,b,c\in {\mathbb {Q}}\}

and, if ω is one of the two non-real cubic roots of 2, then the map

{\begin{array}{rccc}\sigma :&{\mathbb {Q}}({\sqrt[ {3}]{2}})&\longrightarrow &{\mathbb {A}}\\&a+b{\sqrt[ {3}]{2}}+c{\sqrt[ {3}]{4}}&\mapsto &a+b\omega {\sqrt[ {3}]{2}}+c\omega ^{2}{\sqrt[ {3}]{4}}\end{array}}

is an embedding of {\mathbb {Q}}({\sqrt[ {3}]{2}}) in {\mathbb {A}} whose restriction to {\mathbb {Q}} is the identity. However, σ is not an automorphism of {\mathbb {Q}}({\sqrt[ {3}]{2}}).

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

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 (i.e., L ⊃ E ⊃ K), then L is a normal extension of E.
  • 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.

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 which is minimal, i.e. such that the only subfield of M which contains L and which 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

References

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