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Pries 605A Algebraic Number Theory spring 2016. Homework 3. Due Wed 2/10. 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 . √ √ √ √ 14), and L = Q( 6, 14) 3. Let L1 = Q( 6), L2√= Q( √ √ = L1 L2 . √ Show that OL 6= Z[ 6, 14] (even though OL1 = Z[ 6] and OL2 = Z[ 14]). √ √ 4. Suppose r ∈ Z is cube-free and α = 3 r. Find the discriminant of L = Q( 3 r) over Q with respect to the basis S = {1, α, α2 }. 5. 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 .