Could someone please give me a sketch of proof (or the proof itself) for the following claim. I'd appreciate it.

The claim is:

If $\displaystyle x$ is algebraic number prove that $\displaystyle \frac{1}{x}$ also is algebraic number.

Now I know a little bit of higher math/algebra and I know that algebraic numbers form a field and thus a quotient of two algebraic numbers is algebraic number. Is there a proof that does not require the axioms of field? Can someone give me an advice (or two)?

Thx in advance.