Talk:Semi-local ring

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${\displaystyle \mathbb {F} _{q}[X]}$ 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 ${\displaystyle \mathbb {F} _{q}}$. Under the action of the absolute Galois group, a chosen x is conjugated only to finitely many other elements of the algebraic closure.