MATH 3231 Class Notes 02-17-2025

notering-theory

Defined Irreducibility Criterion

Defined Eisenstein Irreducibility Criterion

UFDs and Polynomials

Defined Irreducible

Defined Unique Factorization Domain

Remark

$a,b,c,d\in \mathbb{Z}$ (or $\mathbb{R}$) $a+bi\sqrt{3}=c+di\sqrt{3}⟺a=c,b=d$ (because $i\sqrt{3}\notin \mathbb{R}$).

Corollary of Remark

$2\ne (a+bi\sqrt{3})(i\pm i\sqrt{3})=a\mp 3b)+(\pm a+b)i\sqrt{3}$. We want $a\mp 3b=2$, and $\pm a+b=0$. Say…