Harvard Math 129: Algebraic Number Theory

Midterm

William Stein

Due: 10:10am in class on Thursday, March 17, 2005

This midterm is worth 25% of your grade. There are four problems and this exam is out of 100 points. The point value of each problem is as indicated.

Rules: You may use your course notes and textbook, but no other books, the internet, or people. You may use math software, including Magma and PARI, though I don’t think you will need to for any of these problems, since they are all conceptual.

1. (20 points) Let R be a noetherian integral domain, and let K = Frac(R) be the field of fractions of R. Let \overline{K} be an algebraic closure of K. Let \overline{R} be the set of α ∈ \overline{K} such that there is a nonzero monic polynomial f(x) ∈ R[x] with f(α) = 0. Is \overline{R} necessarily a ring? Prove or give a counterexample.

2. (10 points) 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 the discriminant of O_{K} is (−1)^{s}.

3. Suppose K is a number field. For any finite extension L of K, define set-theoretic maps

Ψ_{L} : C_{K} → C_{L},  [I] ↦ [IO_{L}]
Φ_{L} : C_{L} → C_{K},  [I] ↦ [I ∩ O_{K}],

where [I] denotes the class of the nonzero integral ideal I.

(a) (10 points) Is Ψ_{L} a group homomorphism? Prove or give a counterexample.

(b) (10 points) Is Φ_{L} a group homomorphism? Prove or give a counterexample.

(c) (20 points) Prove that there is a number field L such that Ψ_{L} is the 0 map, i.e., Ψ_{L} sends every element of C_{K} to the identity of C_{L}. [Hint: Use finiteness of C_{K} in two ways.]

(Note: I wonder, is there always an L such that Φ_{L} is the 0 map? This question just occured to me while writing this exam. If you find an answer and tell me the answer, I’ll be very thankful, though this is not part of the official exam. This question is related to “visibility of Mordell-Weil groups”, which I just wrote a paper about.)

4. A number field is totally real if every embedding is real, i.e., s = 0, and a number field is totally complex if every embedding is complex, i.e., r = 0.

(a) (15 points) Find with proof the possible degrees of totally real fields.

(b) (15 points) Find with proof the possible degrees of totally complex fields.