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Section Principal Ideal Domains and Unique Factorization Domains
Worksheet Part 1: A Domain which is not a UFD
In this activity, you’ll show that there exist integral domains which are not unique factorization domains.
1.
In this problem, you will show that
\(\Z[\sqrt{-6}]\) is not a unique factorization domain.
(a)
Find the units in
\(\Z[\sqrt{-6}]\text{.}\)
(b)
Give two factorizations of
\(10\) in
\(\Z[\sqrt{-6}]\) where both factors are not units.
(c)
Use the norm function to show that these factors cannot be paired up where the elements in each pair are associates of each other.
