Irreducible Variety of Irreducible Polynomial Prove that if $f \in k[x_1,...x_n]$ is an irreducible polynomial, then the variety $V(f) \subseteq A^n_k$ is an irreducible variety. 
Basically, I think that I want to prove that the ideal which corresponds to the variety is prime (since there is a bijective correspondence between the two if we are algebraically closed and ideal is radical (which I am assuming, because otherwise, I do not think this will be true)). 
However, I do not think that a principal ideal generated by an irreducible polynomial is necessarily prime. Is that true? And I am I taking this in the right direction?

Since $k[x_1, \ldots, x_n]$ is a UFD, irreducible elements are prime. Thus any principal ideal generated by an irreducible polynomial is in fact a prime ideal, and so the corresponding variety will be irreducible.
