Topology of Numbers · Problem 8.1.7

Quadratic Fields — Allen Hatcher

Mathematics — Number Theory · prime-factorization-quadratic-fields

Problem

(a) Show that if an odd prime

p

factors in

\mathbb{Z}[\omega]

for

\omega = (1 + \sqrt{-3})/2

then it factors in

\mathbb{Z}[\sqrt{-3}]

. (b) Do the same with

-3

replaced by

-7

. (c) Show that this no longer holds when

-3

is replaced by

-11