Maria Agnesi

The fact that "Z_p is a field if and only if p is a prime" can be derived from the fact that "every finite integral domain is a field". (See 69th article.) Also, you can easily derive that every field contains a subfield which is isomorphic to one of Q or Z_p for a prime p. That's why the fields Z_p and Q are called prime field. From the fact that "Z_p is a field if and only if p is a prime", we can derive the Fermat's Little Theorem:

Theorem 1 (Fermat)  If p is a positive prime number then the followings hold.
(1) For all a∈Z, a^p ≡ p (mod p).
(2) if (a, p)=1, that is a and p are relatively primes, then a^p-1 ≡ 1 (mod p).

First Proof of Theorem 1.  For any field, the nonzero elements form a group under the field multiplication. In particular, for Z_p, the elements 1, 2, 3, …, p-1 form a group of order p-1 under multiplication modulo p. Since the order of any element in a group divides the order of the grup, we see that for b≠0 and b∈Z_p, we have b^p-1 =1 in Z_p. Using the fact that Z_p is isomorphic to the ring of cosets of the form a+pZ, we see at once that for any a∈Z not in the coset 0+pZ, we must have a^p-1 ≡ 1 (mod p). It completes the proof of (2) and (1) follows from (2).  QED.

It is used in the proof above that "Z_p is a field if and only if p is a prime" and other properties of ring and coset. In fact, Theorem 1 can be proved not using such priliminaries but using only number theoric facts:

Second Proof of Theorem 1.  First, 0^p ≡ 0 holds. Second we prove (1) by mathematical induction for a. First, it is trivial that 1^p ≡ 1 (mod p). Now assume that n^p ≡ n for n∈N. Then (n+1)^p ≡ n^p + 1^p ≡ n+1 (mod p) holds by properties of modulo. Thus n^p ≡ n for all n∈N. Third, (-n)^p ≡ -n^p ≡ -n (mod p) and it proves (1). (2) follows from (1).  QED.

It is interesting that the proposition "Z_p is a field if and only if p is a prime" can be derived from Theorem 1.

Corollary 2  Z_p is a field if and only if p is a prime.

Proof  Assume that p is a prime. Let n be any element of Z_p and n be not equal to 0 nor to 1. Then n and p are relatively prime, n^p-1 ≡ 1 (mod p) holds, thus n^p-2 is the multiplicative inverse of n.
Next, assume that p is not a prime and p=rs while r and s are integers larger than 1. Then r and s are zero divisors in Z_p thus Z_p is not a field.  QED.

Reference
John B. Fraleigh, A First Course in Abstract Algebra 7ed, Addison-Wesley.

Posted by Maria Agnesi