Math 627
Homework #1
September 10, 2014
Due Wednesday, September 17
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√ 1. (a) Show that the integral closure of Z in Q( 5) is Z 1+2 5 .
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(b) Show that the integral closure of Z in Q( 3 2) is Z[ 3 2].
2. For non-zero n ∈ Z, show that the ring Z n1 is integrally closed.
3. Let K be a number field. Show that there is a basis for K as a Q-vector space
consisting of algebraic integers. (Hint: Start by showing that for any λ ∈ K, there is
some non-zero integer c ∈ Z such that cλ is an algebraic integer.)
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