Pries: 676 Number Theory. 2010. Homework 3.
Integral closure
1. Show Q[x, y]/(y 2 − x3 − x2 ) is not integrally closed.
√
2. Suppose d ∈ √
Z is square-free. If d ≡ 3 mod 4, let Od = Z[ d]. If d ≡
√ 1 mod 4, let
Od = Z[(1 + d)/2]. Prove that Od is the integral closure of Z in Q( d). Find the
discriminant of an integral basis of Od .
√
√
√ √
3. Let L1 = Q( 6), L2√= Q(
√ = L1 L2 .
√
√ 14), and L = Q( 6, 14)
Show that OL 6= Z[ 6, 14] (even though OL1 = Z[ 6] and OL2 = Z[ 14]).
4. Suppose R is an integral domain and consider the algebra QR of quaterions over R.
Here Q is the free R-module with basis {1, i, j, k} and multiplication i2 = j 2 = k 2 = −1,
ij = −ji = k, jk = −kj = i, and ki = −ik = j. If α = a0 + a1 i + a2 j + a3 k, then
let α = a0 − a1 i − a2 j − a3 k. The norm map N : QR → R given by N (α) = αα is
multiplicative. Find a unit of the integral closure of Z in QQ which is not in QZ .