Math 620  Problem Set #11  Due Fri., Dec. 7, 2001

1. Let p be a prime, and let a, b be integers such that a^{2} + b^{2} + 1 ≡ 0 (mod p). Consider the set Λ of points (x, y, z, t) ∈ ℝ^{4} such that x, y, z, t ∈ ℤ and

x ≡ az + bt (mod p), y ≡ bz − at (mod p).

a) Show that Λ is a lattice. Describe it explicitly for p = 3, 5, 7. (In particular, show that a, b must exist, for all p.)

b) Find the volume of the “fundamental parallelotope” spanned by the four “basis vectors”. [Hint: Use a 4 × 4 determinant.]

c) For p = 3, 5, 7, find all the lattice points in the ball B given by x^{2}+y^{2}+z^{2}+t^{2} < 2p.

d) Deduce that there are always lattice points in B other than the origin. [Hint: Use Problem Set 10 #3, and the fact that the volume of a 4-dimensional ball of radius r is \frac{1}{2}π^{2}r^{4}.]

e) In (c), what can you say about the distance of the points to the origin? Make a conjecture and prove it in general.

f) Deduce that every prime p is a sum of four squares. From that, deduce that every positive integer is the sum of four squares.

g) Is every positive integer the sum of three squares?

2. Using your computations in problem 1 above and in Problem Set 8 #4:

a) Write 3, 5, 7 and 105 = 3 · 5 · 7 as sums of four squares.

b) Determine whether 2501 = 41 · 61 can be written as the sum of two squares. If so, do so. If not, write it as the sum of four squares.

c) Redo part (b) for 3569 = 43 · 83.

3. a) Which primes of ℤ are split in O = ℤ[\sqrt{7}]? Which remain prime? Which ramify? find the decomposition groups, inertia groups, and Frobenius automorphisms at the primes of O.

b) What if O is replaced by ℤ[\sqrt{3}, \sqrt{7}]?

4. Which primes (i.e. non-zero prime ideals) of A = ℝ[x] are split in B = A[\sqrt{x}]? Which remain prime? Which ramify? Find the decomposition groups and inertia groups at each prime.

5. a) Using the continued fraction expansion for \sqrt{11}, find an element α ∈ O = ℤ[\sqrt{11}] of norm ±2. Is α ∈ (2)? Is α^{2} ∈ (2)?

b) Use part (a) to find an explicit ideal I ∈ O such that I^{2} = (2). How does this relate to the factorization of the rational prime 2 in O?