Talk:Semi-local ring

$\mathbb {F} _{q}[X]$ is not semilocal

$\mathbb {F} _{q}[X]$ is not semilocal, as the maximal ideals correspond to irreducible polynomials in the ring. There are infintely many of them (all the minimal polynomials of elements x of the alg. closure of $\mathbb {F} _{q}$. Under the action of the absolute Galois group, a chosen x is conjugated only to finitely many other elements of the algebraic closure.