Homework Set 4
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1. Suppose d is a squarefree integer different from 1. Let R = Z[ d].
a) Compute
the matrix for the trace pairing with respect to the basis
√
1, d.
b) Compute the discriminant of this pairing with this basis.
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c) Find the monic irreducible polynomial for d over Q and compute
the discriminant of that to find the discriminant of R over Z.
2. Suppose d is a√ squarefree integer different from 1 and d ≡ 1 (mod 4).
Let R = Z[ 1+2 d ].
a) Compute
the matrix for the trace pairing with respect to the basis
√
1+ d
1, 2 .
b) Compute the discriminant of this pairing with this basis.
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c) Find the monic irreducible polynomial for 1+2 d over Q and compute the discriminant of that to find the discriminant of R over
Z.
3. Suppose d is a √squarefree integer different from 1 and d ≡ 1 (mod 4).
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Let R = Z[ 1+2 d ] and S = Z[ d]. Give two explanations for why
[R : S] = 2 as abelian groups, one from discriminants, and one directly
using basis elements of these free abelian groups.
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