Excerpted from Beachy/Blair, Abstract Algebra, 2nd Ed. © 1996

## § 6.5 Finite fields

Proposition 6.5.1. Let F be a finite field of characteristic p. Then F has pn elements, for some positive integer n.

If F is any field, then the smallest subfield of F that contains the identity element 1 is called the prime subfield of F. If F is a finite field, then its prime subfield is isomorphic to Zp, where p=chr(F) for some prime p.

Theorem 6.5.2. Let F be a finite field with k = pn elements. Then F is the splitting field of the polynomial   xk - x   over the prime subfield of F.

Example 6.5.1. [Wilson's theorem] Let p > 2 be a prime number. Then

(p-1)! -1 (mod p).

Corollary 6.5.3. Two finite fields are isomorphic if and only if they have the same number of elements.

Lemma 6.5.4. Let F be a field of prime characteristic p, let n be a positive integers, and let k = pn. Then

{ a in F | ak = a }

is a subfield of F.

Proposition 6.5.5. Let F be a field with pn elements. Each subfield of F has pm elements for some divisor m of n. Conversely, for each positive divisor m of n there exists a unique subfield of F with pm elements.

Lemma 6.5.6. Let F be a field of characteristic p. If n is a positive integer not divisible by p, then the polynomial   xn - 1   has no repeated roots in any extension field of F.

Theorem 6.5.7. For each prime p and each positive integer n, there exists a field with pn elements.

Definition 6.5.8. Let p be a prime number and let n be a positive integer. The field (unique up to isomorphism) with pn elements is called the Galois field of order pn, denoted by GF(pn).

Lemma 6.5.9. Let G be a finite abelian group. If a is an element of maximal order in G, then the order of every element of G is a divisor of the order of a.

Theorem 6.5.10. The multiplicative group of nonzero elements of a finite field is cyclic.

Theorem 6.5.11. Any finite field is a simple extension of its prime subfield.

Corollary 6.5.12. For each positive integer n there exists an irreducible polynomial of degree n over GF(p).

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