Pries: 676 Number Theory. 2010. Homework 2.
Let p be an odd prime.
Quadratic Reciprocity
1. Suppose p ≡ 3 mod 8 and that q = (p − 1)/2 is also prime. Show that 2 is a primitive
root mod p.
2. (IR 5.11) Suppose p ≡ 3 mod 4 and that q = 2p + 1 is also prime. Prove that 2p − 1 is
not prime.
3. (IR 5.16) Use quadratic reciprocity to find the primes for which 7 is a quadratic residue.
Cyclotomic fields
1. Let w = ζp +ζp−1 . Show that Q(w) is the fixed field of Q(ζp ) under complex conjugation
(more precisely, the fixed field under the unique subgroup of Gal(Q(ζp ) : Q) of order
2). Find a minimal polynomial for w over Q.
2. Given r ∈ N, prove that for some natural number n, there a subfield L ⊂ Q(ζn ) such
that Gal(L : Q) ' Z/r.
Gauss sums
1. * Find a formula for the number of solutions (x, y) with x, y ∈ Fp to the equation
y 2 = x3 + 1 if p ≡ 2 mod 3 and prove it is correct.
2. * (IR 6.8) Let ω = e2πi/3 . If p 6= 3, determine (−3/p) using the Gauss sum method of
Ireland/Rosen 6.2 and the equation (2ω + 1)2 = −3.