1. ## ring of fractions

Consider $\displaystyle \mathbb{Z}[2\sqrt2]=\{a+2b\sqrt{2}|a,b \in \mathbb{Z}\}$, is it's ring of fractions $\displaystyle \{c+2d\sqrt{2}|c,d \in \mathbb{Q}\}$?

2. Originally Posted by dori1123
Consider $\displaystyle \mathbb{Z}[2\sqrt2]=\{a+2b\sqrt{2}|a,b \in \mathbb{Z}\}$, is it's ring of fractions $\displaystyle \{c+2d\sqrt{2}|c,d \in \mathbb{Q}\}$?
You need to show that if $\displaystyle F$ is a field that contains $\displaystyle \mathbb{Z}[2\sqrt{2}]$ then $\displaystyle F$ contains $\displaystyle \{ c+2d\sqrt{2}|c,d\in \mathbb{Q}\}$.

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3. I was wondering for $\displaystyle \mathbb{Z}[2\sqrt{2}]$, is it's ring of fractions $\displaystyle \{ c+2d\sqrt{2}|c,d\in \mathbb{Q}\}$ or $\displaystyle \{\frac{a+2b\sqrt2}{c+2d\sqrt2}|a,b,c,d\in\mathbb{ Z}\}$?

4. Originally Posted by dori1123
I was wondering for $\displaystyle \mathbb{Z}[2\sqrt{2}]$, is it's ring of fractions $\displaystyle \{ c+2d\sqrt{2}|c,d\in \mathbb{Q}\}$ or $\displaystyle \{\frac{a+2b\sqrt2}{c+2d\sqrt2}|a,b,c,d\in\mathbb{ Z}\}$?
I think it is the second one because it contains all elements of the form $\displaystyle \alpha/\beta$ where $\displaystyle \alpha,\beta \in D^{\times}$.
Also, it is not really called "ring of fractions" but "field of fractions".