Abstract Algebra Theory and Applications - Computer Science ...

350 CHAPTER 20 FINITE FIELDSGF(p 24 )✧ ❜✧ ❜GF(p 8 ) GF(p 12 )✧ ✧✧✧GF(p 4 ) GF(p 6 )✧ ✧✧✧GF(p 2 ) GF(p 3 )❜❜ ✧ ✧GF(p)Figure 20.1. Subfields of GF(p 24 )Theorem 20.7 If G is a finite subgroup of F ∗ , the multiplicative group ofnonzero elements of a field F , then G is cyclic.Proof. Let G be a finite subgroup of F ∗ with n = p e 11 · · · pe kkelements,where p i ’s are (not necessarily distinct) primes. By the Fundamental Theoremof Finite Abelian Groups,G ∼ = Z pe 11× · · · × Z pe kk.Let m be the least common multiple of p e 11 , . . . , pe kk. Then G contains anelement of order m. Since every α in G satisfies x r − 1 for some r dividingm, α must also be a root of x m − 1. Since x m − 1 has at most m roots inF , n ≤ m. On the other hand, we know that m ≤ |G|; therefore, m = n.Thus, G contains an element of order n and must be cyclic.□Corollary 20.8 The multiplicative group of all nonzero elements of a finitefield is cyclic.Corollary 20.9 Every finite extension E of a finite field F is a simpleextension of F .Proof. Let α be a generator for the cyclic group E ∗ of nonzero elementsof E. Then E = F (α).□