Topology of Numbers · Problem 8.2.1

Quadratic Fields — Allen Hatcher

Mathematics — Number Theory · unique-factorization-euclidean-algorithm

Problem

(a) According to Proposition 8.10, unique factorization fails in Z[D]\mathbb{Z}[\sqrt{D}] when D=−3D = -3 since the number D(D−1)=12D(D-1) = 12 has two distinct prime factorizations in Z[D]\mathbb{Z}[\sqrt{D}]. On the other hand, if we enlarge Z[−3]\mathbb{Z}[\sqrt{-3}] to Z[ω]\mathbb{Z}[\omega] for ω=(1+−3)/2\omega = (1 + \sqrt{-3})/2 then unique factorization is restored. Explain how the two prime factorizations of 1212 in Z[−3]\mathbb{Z}[\sqrt{-3}] give rise to the same prime factorization in Z[ω]\mathbb{Z}[\omega] (up to units). (b) Do the same thing for the case D=−7D = -7.

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