Homework Assignment 6: Quadratic Forms Due Friday November 8

William Stein

Math 124  Harvard University  Fall

2002

Instructions: Please work with others. (It is possible to do some of the explicit computations in quadratic fields using the number fields functionality of MAGMA or some other computer program; feel free to do so, but describe what you do.) There are 7 problems.

1. (2 points) Let n be a positive integer. Prove that n is a sum of two integer squares if and only if n is a sum of two rational squares.

2. (a) (3 points) Find a positive integer n that has at least three different representations as the sum of two squares, disregarding signs and the order of the summands.

(b) (3 points) Prove that for every r ≥ 1, there is an integer n that has at least r different representations as the sum of two squares, disregarding signs and the order of the summands.

3. Prove the following two statements. You may assume (without proof) that the ring O_{K} of integers in K = ℚ(\sqrt{d}) is a principal ideal domain (and even a euclidean domain) for d = −1, ±2, ±3, +5. Everywhere below x and y are integers.

(a) (2 points) A prime p is of the form x^{2} + 2y^{2} if and only if p = 2 or p ≡ 1 or 3 (mod 8).

(b) (2 points) A prime p is of the form x^{2} − 2y^{2} if and only if p = 2 or p ≡ ±1 (mod 8).

(c) (2 points) Let p be a prime. Then p or −p is of the form x^{2} − 3y^{2} if and only if p = 2, 3 or p ≡ ±1 (mod 12).

4. Assume for the moment that K = ℚ(\sqrt{−5}) is a principal ideal domain (it’s actually not, as you will see below).

(a) (3 points) Prove under this hypothesis that p is of the form x^{2} + 5y^{2} if and only if p = 5 or p ≡ 1, 3, 7, 9 (mod 20).

(b) (2 points) Give an explicit example to show that the statement you proved in part (a) is false.

(c) (1 point) Conclude that ℤ[\sqrt{−5}] is not a principal ideal domain.

5. Let O = ℤ[\sqrt{−5}]. Let I_{1} = (3, 1 + \sqrt{−5}) and I_{2} = (5, \sqrt{−5}).

(a) (3 points) Find two elements α, β ∈ O such that

(α, β) = I_{1}I_{2} = {x_{1}x_{2} : x_{1} ∈ I_{1}, x_{2} ∈ I_{2}}.

(b) (5 points) Prove that every ideal in O can be generated by two elements.

6. (a) Show that there are exactly 3 equivalence classes of positive definite binary quadratic forms of discriminant −23. Find reduced representatives explicitly.

(b) Let α = \frac{1+\sqrt{−23}}{2} and set O = ℤ[α]. By the result of (a), the ideal class group Pic(O) of O has order 3. Find three ideals in O that representative the three distinct elements of Pic(O).

(c) Do the following two ideals define the same equivalence class in Pic(O)?

I_{1} = (4, 3 + 3α), I_{2} = (32, 19 + 27α).

7. (3 points) Let d ≠ 0, 1 be a square-free integer, and let K = ℚ(\sqrt{d}) be the field generated by \sqrt{d}. Let O_{K} be the ring of integers in K (thus O_{K} = ℤ[\frac{1+\sqrt{d}}{2}] if d ≡ 1 (mod 4) and O_{K} = ℤ[\sqrt{d}] otherwise). We say that a prime p ∈ ℤ ramifies in K if the ideal pO_{K} equals ℘^{2} for some nonunit ideal ℘ ⊂ O_{K}. For each of the following d, find the primes p that ramify in K = ℚ(\sqrt{d}):

d = −1, 2, 5, −5, 21, −389.

[Hint: One approach is to consider O_{K}/pO_{K} = O_{K}/℘^{2}O_{K}. Note that the latter ring has elements x ≠ 0 with the property that x^{2} = 0.]