Question

If $R=\mathrm{\mathbb{Z}}\left[\sqrt{d}\right]$, then show that$F\cong \left\{r+s\sqrt{d}/r,s\in \mathrm{\mathbb{Q}}\right\}$ .

Short answer

It is proved that $F\cong \left\{r+s\sqrt{d}/r,s\in \mathrm{\mathbb{Q}}\right\}$.

Step-by-step solution

Use Theorem 10.31

Let R be an Integral domain and F is its field of quotients. If K is field containing R then, K contains a subfield E such that $R⊑S⊑K$and E is isomorphic to F.

Prove that F≅r+sd/r,s∈ℚ

Let $F\cong \left\{r+s\sqrt{d}/r,s\in \mathrm{\mathbb{Q}}\right\}$and $R=\mathrm{\mathbb{Z}}\left[\sqrt{d}\right]$.

Here, F is the field of quotient of R .

Therefore, $R⊑S⊑K$.

Now, by Theorem 10.31, we can say that $F\cong S$.

Therefore, $F\cong \left\{r+s\sqrt{d}/r,s\in \mathrm{\mathbb{Q}}\right\}$.

Hence proved.