Abstract Algebra?

Question

Suppose p is a prime and p is congruent to p (mod 4), (ie. p = 4k +3). How do I show that there is no element a in U(p) such that a^2 = p - 1?

Answer 1

Write the equation as a² + 1 = p with p ≡ 3(mod 4) (*)
If a is even, a² ≡ 0(mod 4) so a² + 1 ≡ 1(mod 4
If a is odd, a² ≡ 1(mod 4), so a² + 1 ≡ 2(mod 4),
consequently, (*) can never hold.