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Abstract Algebra

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Units in a Ring: Finitely Many in $Z[\sqrt{D}]$, Some Small $D$?

ABSALG-SJZLRH

For $D\in\mathbb{Z}$, let $R_D$ be the ring $\mathbb{Z}[\sqrt{D}]=\{m+n\sqrt{D}\mid m,n\in\mathbb{Z}\}$.

Assume throughout this question that $D\not=1$ is square-free, that is, it is not divisible by the square of any prime number in $\mathbb{Z}$.

A unit in a ring $R$ with identity $1$ is an element $u\in R$ with a multiplicative inverse: $uv=1$, for some $v\in R$.

True

False

True

False

$\mathbb{Z}$ has only finitely many units.

True

False

$\mathbb{Z}[\sqrt{-1}]$ has only finitely many units.

True

False

$\mathbb{Z}[\sqrt{2}]$ has only finitely many units.

True

False

The units in $\mathbb{Z}[\sqrt{D}]$ are the integer solutions of $a^2+|D|b^2=\pm1$.