Math 129: Algebraic Number Theory Homework Assignment 4

William Stein

Due: Thursday, March 11, 2004

The problems:

1. Find representative ideals for each element of the class group of Q(\sqrt{−23}). Illustrate how to use the Minkowski bound to prove that your list of representatives is complete.

2. Suppose O is an order in the ring of integers O_{K} of a number field. Is every ideal in O necessarily generated by two elements?

3. Let K be a number field of degree n > 1 with s pairs of complex conjugate embeddings. Prove that

(\frac{π}{4} )^{s} \frac{n^{n}}{n!} > 1.

4. Do the exercise on page 19 of Swinnerton-Dyer, which shows that the quantity C_{r,s} in the finiteness of class group theorem can be taken to be (\frac{4}{π} )^{s} \frac{n!}{n^{n}}.

5. Let α denote a root of x^{3} − x + 2 and let K = Q(α). Show that O_{K} = Z[α] and that K has class number 1 (don’t just read this off from teh output of the MAGMA MaximalOrder and ClassNumber commands). [Hint: consider the square factors of the discriminant of x^{3} − x + 2 and show that \frac{1}{2}(a + bα + cα^{2}) is an algebra integer if and only if a, b, and c are all even.]

6. If S is a closed, bounded, convex, symmetric set in R^{n} with Vol(S) ≥ m2^{n}, for some positive integer m, show that S contains at least 2m nonzero points in Z^{n}.