9.4. MORE ON ALGEBRAIC FIELD EXTENSIONS 223

9.4.4 Separable PolynomialsThis is a good time to make a very important observation about irreducible polynomials.

Lemma 9.4.24 Suppose q(x) ̸= p(x) are both irreducible polynomials over a field F. Thenthere is no root common to both p(x) and q(x).

Proof: If l (x) is a monic polynomial which divides them both, then l (x) must equal1. Otherwise, it would equal p(x) and q(x) which would require these two to be equal.Thus p(x) and q(x) are relatively prime and there exist polynomials a(x) ,b(x) havingcoefficients in F such that

a(x) p(x)+b(x)q(x) = 1

Now if p(x) and q(x) share a root r, then (x− r) divides both sides of the above in K [x]where K is a field which contains all roots of both polynomials. But this is impossible. ■

Now here is an important definition of a class of polynomials which yield equality inthe inequality of Theorem 9.4.12. We know that if p(x) of this theorem has distinct roots,then equality holds. However, there is a more general kind of polynomial which also givesequality.

Definition 9.4.25 Let p(x) be a polynomial having coefficients in a field F. Also let K bea splitting field. Then p(x) is separable if it is of the form

p(x) =m

∏i=1

qi (x)ki

where each qi (x) is irreducible over F and each qi (x) has distinct roots in K. From theabove lemma, no two qi (x) share a root. Thus

p1 (x)≡m

∏i=1

qi (x)

has distinct roots in K.

Example 9.4.26 For example, consider the case where F=Q and the polynomial is of theform (

x2 +1)2 (

x2−2)2

= x8−2x6−3x4 +4x2 +4

Then let K be the splitting field over Q, Q[i,√

2].The polynomials x2 + 1 and x2− 2 are

irreducible over Q and each has distinct roots in K.

Then the following corollary is the reason why separable polynomials are so important.Also, one can show that if F contains a field which is isomorphic to Q then every poly-nomial with coefficients in F is separable. This will be done later after presenting the bigresults. This is equivalent to saying that the field has characteristic zero. In addition, theproperty of being separable holds in other situations.

Corollary 9.4.27 Let K be a splitting field of p(x) over the field F. Assume p(x) isseparable. Then

|G(K,F)|= [K : F]