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[edit]

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[edit]

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[edit]

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[edit]

References[edit]