9.4. MORE ON ALGEBRAIC FIELD EXTENSIONS 221

9.4.2 Normal Field ExtensionsThe following is the definition of a normal field extension.

Definition 9.4.18 Let K be a finite dimensional extension of a field F such that every el-ement of K is algebraic over F, that is, each element of K is a root of some polynomialin F [x]. Then K is called a normal extension if for every k ∈ K all roots of the minimumpolynomial of k are contained in K.

So what are some ways to tell that a field is a normal extension? It turns out that if K isa splitting field of f (x) ∈ F [x] , then K is a normal extension. I found this in [26]. This isan amazing result.

Proposition 9.4.19 The following are valid

1. Let K be a splitting field of f (x) ∈ F [x]. Then K is a normal extension.

2. If L is an intermediate field between F and K where K is a normal field extension ofF, then L is also a normal extension of F.

Proof: 1.) Let r ∈ K ≡ F(a1, ...,aq) where{

a1, ...,aq}

are the roots of f (x) and letg(x) be the minimum polynomial of r with coefficients in F. Thus, g(x) is an irreduciblemonic polynomial in F [x] having r as a root. It is required to show that every other root ofg(x) is in K. Let the roots of g(x) in a splitting field be {r1 = r,r2, · · · ,rm}. Now g(x) isthe minimum polynomial of r j over F because g(x) is irreducible by Lemma 9.4.7.

By Theorem 9.4.5, there exists an isomorphism η of F(r1) and F(r j) which fixes Fand maps r1 to r j. Thus η is an extension of the identity on F. Now K(r1) and K(r j) aresplitting fields of f (x) over F(r1) and F(r j) respectively. By Theorem 9.4.9, the two fieldsK(r1) and K(r j) are isomorphic, the isomorphism, ζ extending η . Hence

[K(r1) :K] = [K(r j) :K]

But r1 ∈ K and so K(r1) = K. Therefore, [K(r j) :K] = 1 and so K = K(r j) and so r j isalso in K. Thus all the roots of g(x) are in K.

2.) Consider the last assertion. Suppose r = r1 ∈ L where the minimum polynomialfor r is denoted by q(x). Then since K is a normal extension, all the roots of q(x) are in K.Let them be {r1, · · · ,rm}. By Theorem 9.4.5 applied to the identity map on L, there existsan isomorphism θ : L(r1)→ L(r j) which fixes L and takes r1 to r j. But this implies that

1 = [L(r1) : L] = [L(r j) : L]

Hence r j ∈ L also. If r j /∈ L, then{

1,r j}

is independent and so the dimension would be atleast 2. Since r was an arbitrary element of L, this shows that L is normal. ■

9.4.3 Normal Subgroups and Quotient GroupsWhen you look at groups, one of the first things to consider is the notion of a normalsubgroup. The word “normal” is greatly over used in math. Its meaning in this context isgiven next.