$I=(f_1, \ldots, f_n)\subset k[x_1, \ldots, x_n]$ with $f_i\in k[x_i]$ irreducible polynomials

Question

Let $A=k[x_1,\ldots, x_n]$ and $I=(f_1, \ldots, f_n)\subset A$ with $f_i\in k[x_i]$ irreducible polynomials. Is it true that $I$ is a maximal ideal in $A$?

$I$ is a maximal ideal $\iff$ $1\in (I, g)$ for every $g\in A\setminus I$ $\iff$ $1\in (I, g)$ for every $g\in A$ with $\deg g<\deg f_i$ $\forall i$ (with the the lexicographical order). Any other ideas?

Answer 1

Of course not! $A/I\simeq k[x_1]/(f_1)\otimes_k\cdots\otimes_kk[x_n]/(f_n)$, and a tensor product of fields is not necessarily a field. For a concrete (counter)example see this answer.