SUMMARY
The discussion focuses on proving that the ring $\mathbb{Z}[\sqrt{-3}]$ is not a Euclidean domain by identifying an irreducible element that is not prime. The user proposes the element 2 as irreducible, as its norm N(2) equals 4, which leads to the conclusion that any factorization involving 2 does not yield prime factors. Specifically, 2 divides 4, which can be expressed as the product of the factors (1 + √(-3))(1 - √(-3)), yet does not divide either factor, confirming its status as irreducible but not prime.
PREREQUISITES
- Understanding of Euclidean domains and their properties
- Familiarity with the concept of irreducibility in ring theory
- Knowledge of norms in algebraic number theory
- Basic experience with algebraic integers, specifically in the context of $\mathbb{Z}[\sqrt{-3}]$
NEXT STEPS
- Study the properties of Euclidean domains and their implications in algebraic structures
- Explore the concept of irreducibility versus primality in more depth
- Learn about norms in algebraic number fields and their applications
- Investigate other examples of rings that are not Euclidean domains
USEFUL FOR
This discussion is beneficial for mathematicians, particularly those specializing in algebraic number theory, as well as students and researchers interested in the properties of rings and domains.