I am reading Silverman's Arithmetic of Elliptic Curves, and his early examples of (affine algebraic) varieties are confusing me. This is because he defines algebraic varieties as algebraic sets $V$ that satisfy the condition that $I(V)$ be a prime ideal in $\overline{K}[X]$, but he fails to check whether or not this condition holds in his examples. What is the obvious condition he is using to check whether or not these algebraic sets are varieties?
My guess is that every algebraic set generated by a single irreducible polynomial over $\overline{K}$ (as all of his examples are) will be a variety. To me this isn't obvious though, because for general rings the implication "irriducible element => prime element" does not hold. What's going on here?