Normal extension - Alchetron, The Free Social Encyclopedia

In abstract algebra, an algebraic field extension L/K is said to be normal if every irreducible polynomial, either has no root in L or splits into linear factors in L. Bourbaki calls such an extension a quasi-Galois extension.

Definition

The algebraic field extension L/F is normal (we also say that L is normal over F) if every irreducible polynomial over F that has at least one root in L splits over L. In other words, if α ∈ L, then all conjugates of α over F (i.e., all roots of the minimal polynomial of α over F) belong to L.

Equivalent properties

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

Every embedding σ of L in K^a that restricts to the identity on K, satisfies σ(L) = L (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.)

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.

Examples and Counterexamples

For example, Q ( 2 ) is a normal extension of Q , since it is a splitting field of x² − 2. On the other hand, Q ( 2 3 ) is not a normal extension of Q since the irreducible polynomial x³ − 2 has one root in it (namely, 2 3 ), but not all of them (it does not have the non-real cubic roots of 2). Recall that the field Q ¯ of algebraic numbers is the algebraic closure of Q , i.e., it contains Q ( 2 3 ) . Since, Q ( 2 3 ) = { a + b 2 3 + c 4 3 ∈ Q ¯ | a , b , c ∈ Q } and, if ω is a primitive cubic root of unity, then the map

is an embedding of Q ( 2 3 ) in Q ¯ whose restriction to Q is the identity. However, σ is not an automorphism of Q ( 2 3 ) .

For any prime p, the extension Q ( 2 p , ζ p ) is normal of degree p(p − 1). It is a splitting field of x^p − 2. Here ζ p denotes any pth primitive root of unity. The field Q ( 2 3 , ζ 3 ) is the normal closure (see below) of Q ( 2 3 ) .

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., 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.

References

Normal extension Wikipedia

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