Harvard Math 129: Algebraic Number Theory Homework Assignment 6

William Stein

Due: TUESDAY, March 22, 2005

Note the unusual due date, which is to avoid making the homework due right before spring break.

The problems have equal point value, and multi-part problems are of the same value. In any problem where you use a computer, include in your solution the exact commands you type and their output. You may use any software, including (but not limited to) MAGMA and PARI.

For a number field K, let U_{K} denote the group O_{K}^{∗} of units of its ring of integers. In each of the following problems, answer the question and give a proof that your answer is correct.

1. Let K vary through all number fields. What torsion subgroups (U_{K})_{tor} actually occur?

2. If U_{K} ≈ ℤ^{n} × (U_{K})_{tor}, we say that U_{K} has rank n. Let K vary through all number fields. What ranks actually occur?

3. Let K vary through all number fields such that the group U_{K} of units of K is a finite group. What finite groups U_{K} actually occur?

4. Let K = ℚ(ζ_{5}).

(a) Show that r = 0 and s = 2.

(b) Find explicit generators for the group of units of U_{K}.

(c) Draw an illustration of the log map φ : U_{K} → ℝ^{2}, including the hyperplane x_{1} + x_{2} = 0 and the lattice in the hyperplane spanned by the image of U_{K}.