0
votes
1answer
17 views

Natural norm for the ring $\{a+b\sqrt{2}$ | $a,b \in \mathbb Z \} $?

I am working on showing that $\{a+b\sqrt{2}$ | $a,b \in \mathbb Z \} $ is an euclidean domain. There was a similar problem showing that $\{a+b\sqrt{-2}$ | $a,b \in \mathbb Z \} $ was an euclidean ...
2
votes
2answers
72 views

Generalizing norms for modules

An normed space is defined as a vector space V plus a norm operation over $V$. Is it meaningful to generalize this notion to modules, where one is dealing with rings instead of fields? What I'm ...
0
votes
2answers
60 views

On Real Hamilton Ring ..

i know the definition of real hamilton ring but if we said ,$ I$ is the ring of integral hamilton what does this mean ? what is the properites that word , integral , adds to the structure of ...
2
votes
2answers
182 views

Does $N(z)=\pm 1$ imply $z$ is a unit in $\mathbb{Z}[\sqrt{10}]$?

I've been trying to prove that $\mathbb{Z}[\sqrt{10}]$ is not factorial. I did this by defining the norm $N(a+b\sqrt{10})=a^2-10b^2$. I was able to show for myself that $N(z)=\pm 2$ and $N(z)=\pm 5$ ...
2
votes
4answers
596 views

Prove that $N(\gamma) = 1$ if, and only if, $\gamma$ is a unit in the ring $\mathbb{Z}[\sqrt{n}]$

Prove that $N(\gamma) = 1$ if, and only if, $\gamma$ is a unit in the ring $\mathbb{Z}[\sqrt{n}]$ Where $N$ is the norm function that maps $\gamma = a+b\sqrt{n} \mapsto \left | a^2-nb^2 \right |$ I ...