Abstract Algebra Theory and Applications - Computer Science ...

21.1 FIELD AUTOMORPHISMS 369of degree n is separable if it has n distinct roots in its splitting field E.Equivalently, f(x) is separable if it factors into distinct linear factors overE[x]. An extension E of F is a separable extension of F if every elementin E is the root of a separable polynomial in F [x]. Also recall that f(x) isseparable if and only if gcd(f(x), f ′ (x)) = 1 (Lemma 20.4).Proposition 21.7 Let f(x) be an irreducible polynomial over F [x]. If thecharacteristic of F is 0, then f(x) is separable. If the characteristic of F isp and f(x) ≠ g(x p ) for some g(x) in F [x], then f(x) is also separable.Proof. First assume that charF = 0. Since deg f ′ (x) < deg f(x) andf(x) is irreducible, the only way gcd(f(x), f ′ (x)) ≠ 1 is if f ′ (x) is the zeropolynomial; however, this is impossible in a field of characteristic zero. IfcharF = p, then f ′ (x) can be the zero polynomial if every coefficient of f(x)is a multiple of p. This can happen only if we have a polynomial of the formf(x) = a 0 + a 1 x p + a 2 x 2p + · · · + a n x np .□Certainly extensions of a field F of the form F (α) are some of the easiestto study and understand. Given a field extension E of F , the obviousquestion to ask is when it is possible to find an element α ∈ E such thatE = F (α). In this case, α is called a primitive element. We already knowthat primitive elements exist for certain extensions. For example,Q( √ 3, √ 5 ) = Q( √ 3 + √ 5 )andQ( 3√ 5, √ 5 i) = Q( 6√ 5 i).Corollary 20.9 tells us that there exists a primitive element for any finiteextension of a finite field. The next theorem tells us that we can often finda primitive element.Theorem 21.8 (Primitive Element Theorem) Let E be a finite separableextension of a field F . Then there exists an α ∈ E such that E = F (α).Proof. We already know that there is no problem if F is a finite field.Suppose that E is a finite extension of an infinite field. We will prove theresult for F (α, β). The general case easily follows when we use mathematicalinduction. Let f(x) and g(x) be the minimal polynomials of α and β,respectively. Let K be the field in which both f(x) and g(x) split. Supposethat f(x) has zeros α = α 1 , . . . , α n in K and g(x) has zeros β = β 1 , . . . , β m