# Thread: Determine if this Ring is a field

1. ## Determine if this Ring is a field

Can you help me prove that this ring is a field?

R = Q[$\displaystyle \sqrt(d)$] = { a+b$\displaystyle \sqrt(d)$ ; a,b are rational}

So I tried:

We must show that every element of R is a unit. So for some element (a+b*$\displaystyle \sqrt(d)$) there must be some (e+c*$\displaystyle \sqrt(d)$) such that

(a+b*$\displaystyle \sqrt(d)$)*(e+c*$\displaystyle \sqrt(d)$) = 1

so (ae+bed) + (be + ac)*$\displaystyle \sqrt(d)$ = 1

so ae-bed = 1 and be+ac =0

but then I get stuck after that

2. I believe that the first equation should be ae+bcd=1. This is just a system of equations in the unknowns e and c which you can solve. Just be careful of the special cases a=0 and possibly b=0.

3. If $\displaystyle d\geq 0$ is a perfect square then $\displaystyle \mathbb{Q}[\sqrt{d}]=\mathbb{Q}$ and so $\displaystyle \mathbb{Q}[\sqrt{d}]$ is a field.

If $\displaystyle d\geq 0$ is not a perfect square, then, $\displaystyle \sqrt{d}$ is irrational.

In this case choose $\displaystyle a+b\sqrt{d}\neq 0$. Then

$\displaystyle (a+b\sqrt{d})(x+y\sqrt{d})=1\Leftrightarrow \begin{Bmatrix} ax+bdy=1\\bx+ay=0\end{matrix}$

Now, using that $\displaystyle \sqrt{d}$ is irrational, prove that

$\displaystyle \det \begin{bmatrix}{a}&{bd}\\{b}&{a}\end{bmatrix}=a^2-b^2d\neq 0$

which implies that the system has a solution $\displaystyle (x,y)=\ldots$ ( besides, unique ).

Fernando Revilla

4. Thanks guys I have it figured out