Math 514a Algebraic Number Theory Fall 1995

Homework assignment 3: Exercises in Marcus referred to below are after Chapter 2.

11. For the purposes of this problem, an order in a number field is a subring which is finitely generated as a Z-module. Prove that an order as defined in class is an order in this sense, but not conversely. Make sense of and prove the statement “the ring of integers is the maximal order.”

12. Prove that if D is the discriminant of a number field, then D ≡ 0, 1 (mod 4). This is “Stickelberger’s criterion.” (See Marcus, exercise 22 for hints.)

13. Prove that if F ⊆ K ⊆ L are number fields, then Tr_{L/F} = Tr_{K/F} Tr_{L/K} and N_{L/F} = N_{K/F} N_{L/K}.

14. Prove that Q(\sqrt{a}) and Q(\sqrt{b}) are isomorphic if and only if a/b is the square of a rational number.

15. Prove that an algebraic integer α is a root of unity (i.e., α^{n} = 1 for some n) if and only if |σ(α)| = 1 for every embedding of Q(α) into C. (See Marcus exercise 11 for hints.) Can you find a counterexample when α is only assumed to be an algebraic number?

16. Exercise 30 of Marcus gives an example of a number field whose ring of integers is not generated by a single element. Do this exercise.

17. This is the first in what will be an ongoing series of exercises:

A “pure cubic” field is one of the form Q( ^{3} \sqrt{a}) with a rational and not a cube. A “biquadratic” field is one of the form Q(\sqrt{b}, \sqrt{c}) with b and c rational with b/c not a square. The class will be divided into the B (biquadratic) team and the C (cubic) team. The exercise this time is to find the ring of integers and discriminant of the corresponding type of field. Exercises 28, 29, 41, and 42 in Marcus outline solutions, although his division into cases may not be the most elegant one.