Pries 605A Algebraic Number Theory spring 2016.
Homework 1. Due Wed 1/27.
(By Friday 1/22) Read handout on quadratic reciprocity by Pete Clark.
Try to get SAGE functioning on some computer you have access to.
Hand in the problems below: those labeled ** or fun are optional.
1. If p is an odd prime, we proved that 2 is a square mod p iff p ≡ ±1 mod 8. Prove the
following are equivalent.
(a) p ≡ ±1 mod 8.
(b) p = x2 − 2y 2 has a solution for some x, y ∈ Z.
√
(c) p factors in Z[ 2].
2. Let p ≥ 5 be a prime. Note the algebraic fact that (2x + 1)2 = −3 iff x3 = 1 and x 6= 1.
Prove that p ≡ 1 mod 3 if and only if x2 ≡ −3 mod p has a solution.
(a) Method 1: using the fact that (Z/p)∗ is cyclic.
(b) Method 2: using the statement of quadratic reciprocity.
(c) ** (IR 6.8) Method 3: using the Gauss sum method of Ireland/Rosen 6.2 (let
ω = e2πi/3 ).
3. (IR 5.16) Use quadratic reciprocity to find the primes for which 7 is a quadratic residue.
4. Suppose p ≡ 3 mod 8 and that q = (p − 1)/2 is also prime. Show that 2 is a primitive
root mod p (a generator of (Z/p)∗ ).
5. An integer a such that 1 ≤ a ≤ p − 1 is a cubic residue if a is in the image of the map
(Z/p)∗ → (Z/p)∗ taking x 7→ x3 . How many cubic residues are there for each prime p?
6. (fun) Find a formula for the number of solutions (x, y) with x, y ∈ Z/p to the equation
y 2 = x3 + 1 if p ≡ 2 mod 3 and prove it is correct.
7. (fun) (IR 5.11) Suppose p ≡ 3 mod 4 and that q = 2p + 1 is also prime. Prove that
2p − 1 is not prime.