Abstract Algebra Theory and Applications - Computer Science ...

20.1 STRUCTURE OF A FINITE FIELD 347φ must be a subfield of F isomorphic to Z p . We will denote this subfield byK. Since F is a finite field, it must be a finite extension of K and, therefore,an algebraic extension of K. Suppose that [F : K] = n is the dimension ofF , where F is a K vector space. There must exist elements α 1 , . . . , α n ∈ Fsuch that any element α in F can be written uniquely in the formα = a 1 α 1 + · · · + a n α n ,where the a i ’s are in K. Since there are p elements in K, there are p npossible linear combinations of the α i ’s. Therefore, the order of F must bep n .□Lemma 20.3 (Freshman’s Dream) Let p be prime and D be an integraldomain of characteristic p. Thenfor all positive integers n.a pn + b pn = (a + b) pnProof. We will prove this lemma using mathematical induction on n. Wecan use the binomial formula (see Chapter 1, Example 3) to verify the casefor n = 1; that is,p∑( )(a + b) p p= a k b p−k .kIf 0 < k < p, then( pkk=0)=p!k!(p − k)!must be divisible by p, since p cannot divide k!(p − k)!. Note that D is anintegral domain of characteristic p, so all but the first and last terms in thesum must be zero. Therefore, (a + b) p = a p + b p .Now suppose that the result holds for all k, where 1 ≤ k ≤ n. By theinduction hypothesis,(a + b) pn+1 = ((a + b) p ) pn = (a p + b p ) pn = (a p ) pn + (b p ) pn = a pn+1 + b pn+1 .Therefore, the lemma is true for n + 1 and the proof is complete.Let F be a field. A polynomial f(x) ∈ F [x] of degree n is separable ifit has n distinct roots in the splitting field of f(x); that is, f(x) is separablewhen it factors into distinct linear factors over the splitting field of F . An□