Math 254A. Homework 2

Due Wednesday, 6 September

1. Let A ⊆ B be commutative rings, with B integral over A. Show that B^{∗} ∩ A = A^{∗}.

2. Exercise 7 on page 15: The discriminant d_{K} of an algebraic number field K is always

≡ 0 (mod 4) or ≡ 1 (mod 4) (Stickelberger’s discriminant relation).

Hint: The determinant det(σ_{i}ω_{j}) of an integral basis ω_{j} is a sum of terms, each prefixed by a positive or a negative sign. Writing P , resp. N , for the sum of the positive, resp. negative terms, one finds d_{K} = (P − N)^{2} = (P + N)^{2} − 4P N .