Algebraic number theory M 3/4 P 15

Question Sheet 4

Question 1 of this sheet forms a part of assessed work, due on the 17th of March.

You can use the following theorem proved in the lectures (one of the main results of the course): in every class of ideals of A(d) there exists an integral ideal I with norm ||I|| < λ(d), where

λ(d) = \sqrt{d} if d > 0 is congruent to 2 or 3 modulo 4, λ(d) = \sqrt{d}/2 if d > 0 is congruent to 1 modulo 4, λ(d) = 4\sqrt{|d|}/π if d < 0 is congruent to 2 or 3 modulo 4,

λ(d) = 2\sqrt{|d|}/π if d < 0 is congruent to 1 modulo 4. Also you can use the fact that the norm of the principal ideal generated

by a in A(d) is |N(a)|.

1. Compute the class groups of A(5) and A(6).

2. (non-assessed) Compute the class group of A(−163). [Hint: prove that all ideals of small norms are principal.]

3. (non-assessed) Prove that the class number of A(d) is not bounded.

4. (non-assessed) Prove that the class group of A(−21) is the product of two cyclic groups of order 2. [Hint: Show that the prime ideals over 2 and 3 are not principal, and their product is not principal either.]

5. (non-assessed) Compute the class groups of other quadratic fields of your choice.