Prove that . Conclude that . Deduce Wilson’s Theorem: if is an odd prime, then mod .
Recall that by definition, and that is merely the root having minimal polynomial . So . Comparing constant coefficients, we have , so that $latex , and hence .
Restrict now to the field with odd. Then , and . Thus mod .
(I’d like to point out that this is a really roundabout way to prove Wilson’s Theorem. The easy(ier) way is to note that in , every element but -1 is distinct from its inverse.)