Math 129: Algebraic Number Theory Homework Assignment 3

William Stein

Due: Thursday, March 4, 2004

The problems:

1. (a) Suppose K is a number field of degree 2. Prove that O_{K} = Z[a] for some

a ∈ O_{K}.

(b) Prove that if K and K′ are two number fields of degree 2 and Disc(O_{K}) = Disc(O_{K′}) then K = K′.

2. (*) Does there exist a number field K of degree 4 such that O_{K} ≠ Z[a] for all a ∈ O_{K}? If so, give an explicit example.

3. Let K be the quintic number field generated by a root of x^{5} +7x^{4} +3x^{2} −x+1. Draw a diagram (be creative) that illustrates the factorization of every prime p ∈ Z, with p < 100, in O_{K}.

4. (Problem 1.9 in Swinnerton-Dyer) Show that the only solutions x, y ∈ Z to y^{2} = x^{3} − 13 are given by x = 17, y = ±70, as follows. Factor the equation y^{2} + 13 = x^{3} in the number field Q(\sqrt{−13}), which has class number 2. Show that if x, y is an integer solution then the ideal (y + \sqrt{−13}) must be the cube of an ideal, and hence y + \sqrt{−13} = (a + b\sqrt{−13})^{3}; thus 1 = b(3a^{2} − 13b^{2}).

5. Suppose I and J are ideals in the ring O_{K} of integers of a number field K. Does IJ = I ∩ J? Prove or give a counterexample.

6. Let O_{K} be the ring of integers Q(\sqrt{5}), and let

I = (5, 2 + \sqrt{5})  and  J = (209, (389 + \sqrt{5})/2)

be integral ideals of O_{K}.

(a) Find an element of O_{K} that is congruent to \sqrt{5} modulo I and is congruent to 1 − \sqrt{5} modulo J.

(b) What is the cardinality of (O_{K}/I) ⊕ (O_{K}/J)?

(c) Find an element a ∈ I such that (a)/I is coprime to J.

7. Let O_{K} be the ring of integers of a number field K, and suppose K has exactly 2s complex embeddings. Prove that the sign of Disc(O_{K}) is (−1)^{s}.