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**Proposition 6.5.1.**
Let F be a finite field of characteristic p. Then F has
p^{n} 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
**Z**_{p},
where p=chr(F) for some prime p.

**Theorem 6.5.2.**
Let F be a finite field with k = p^{n} elements.
Then F is the splitting field of the polynomial
x^{k} - 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 = p^{n}. Then

{ a in F | a^{k} = a }

**Proposition 6.5.5.**
Let F be a field with p^{n} elements.
Each subfield of F has p^{m}
elements for some divisor m of n.
Conversely, for each positive divisor m of n there
exists a unique subfield of F with p^{m} 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 x^{n} - 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 p^{n} 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 p^{n}
elements is called the
**Galois field of order p**^{n},
denoted by GF(p^{n}).

**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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