Math 514 homework #25 Due date: 19951122 D. J. Bernstein 19951115 Let p be an odd prime number. For any integer a, the quantity a^{(p−1)/2} is congruent to either 0, 1, or −1 modulo p. The Legendre symbol (\frac{a}{p}) is defined as 0, 1, or −1 respectively.

In these exercises p is an odd prime number, ζ is a primitive pth root of unity, and G = ∑_{0<a<p} (\frac{a}{p}) ζ^{a}. (G is a Gauss sum.)

Exercise 78. Prove that G^{2} = (\frac{−1}{p}) p.

Exercise 79. Let d be an integer. Prove that Q(\sqrt{d}) ⊆ Q(ζ_{8d}).

Exercise 80. Assume that (\frac{−1}{p}) = 1. Prove that (p^{p} − 1)/(p − 1) is not a prime number. (Hint: Consider (G − ζ^{(p+1)/2})(G + ζ^{(p+1)/2}) in the field Q(ζ).)