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Let $K$ be an arbitrary field. I want to prove that the maximal ideal of $K[x,y]$ is generated by two elements.

If $K$ is algebraically closed, maximal ideals are of the form $（x-a,y-b）$ because of Hilbert's Nullstellensatz. But if $K$ is not algebraically closed, I cannot specify the form of maximal ideals, so I cannot go further.Thank you for your help.

Pont
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    Sketch: Let $m$ be a maximal ideal. Then there is an irreducible $g(x) \in m$ (ie $g(x)$ generates $m \cap K[x]$). Let $K’=K[x]/(g)$, then $m$ corresponds to a maximal ideal of $K’[y]$ so is of the form $(f)$ for $f \in K’[y]$, so in other words $m/(g)$ is principal. Thus $m$ has two generators. – Aphelli Jan 16 '21 at 11:59
  • There is an exercise in the French book Francinou and Gianella "Exercices de mathématiques pour l'agrégation: Algèbre 1", giving a generalization this result. In all generality, one may prove that prime ideals of $K[X,Y]$ are of exactly one of the three following form : the zero ideal ; a principal ideal generated by an irreducible polynomial (this is never maximal) ; an ideal generated by two polynomials P(X) and Q(X,Y) with P irreducible, and Q irreducible in $\left(K[X]/P(X)\right)[Y]$ (they are the maximal ones) – Suzet Jan 16 '21 at 11:59
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    This is a copy of this question https://math.stackexchange.com/questions/56916/what-do-prime-ideals-in-kx-y-look-like – Mo145 Jan 16 '21 at 12:04

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