Is it a field? set of real numbers in the form...

Hi,

**problem:**

Let $\displaystyle Q(\sqrt{2})$ be the set of all real numbers of the form

$\displaystyle \alpha+\beta\sqrt{2}$, where $\displaystyle \alpha\;and\;\beta$ are rational.

(a) Is $\displaystyle Q(\sqrt{2})$ a field?

(b) What if $\displaystyle \alpha\;and\;\beta$ are required to be integers?

attempt:

(a)

I look at the axioms for a field and check whether any of them fail. I am unable to find one that does.

Is it ok to say that I can express any real number in the form $\displaystyle \alpha+\beta\sqrt{2}$ as long as $\displaystyle \alpha\;and\;\beta$ are rational?

(b)

I am unable to find an axiom that "fails", but say I want to write the real number $\displaystyle \frac{\sqrt{2}}{2}$. I don't think I can do this if $\displaystyle \alpha\;and\;\beta$ are integers..

Any comments are very welcome!

Thanks