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Abstract Algebra

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Moderate

Unique Factorization: Is $Z[\sqrt{D}]$ a UFD when $D=1$ mod $4$?

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Let $D\not=0,1$ be a square-free integer, meaning there is no integer $m\not=1$ such that $m^2$ divides $D$.

Let $\mathbb{Z}[\sqrt{D}]:=\{a+b\sqrt{D}\mid a, b\in\mathbb{Z}\}$ be the ring generated over $\mathbb{Z}$ by $\sqrt{D}$.

In the following, decide whether each statement is always true or always false.

Statement Always True

Statement Always False

Always True

Always False

$2$ divides $(1+\sqrt{D})(1-\sqrt{D})$

Always True

Always False

$2$ is prime in $\mathbb{Z}[\sqrt{D}]$

Always True

Always False

$2$ is irreducible in $\mathbb{Z}[\sqrt{D}]$

Always True

Always False

$\mathbb{Z}[\sqrt{D}]$ is a Unique Factorization Domain (UFD)