Please help. Is an ideal $\langle x,y \rangle$ in $k[x,y]$, where $k$ is a field, a prime ideal? I can't prove it. Probably it should be shown that $k[x,y]/\langle x,y \rangle$ is an integral domain, but I don't know what it is isomorphic with.
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Łukasz Matysiak
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3See https://math.stackexchange.com/questions/56916/whatdoprimeidealsinkxylooklike – Michael Morrow Jul 18 '21 at 22:46

hint: consider the map $k[x,y]\to k$ defined by $p(x,y)\mapsto p(0,0)$. what are the kernel and the image of this map? – Atticus Stonestrom Jul 19 '21 at 01:47

Alternatively, it's easy to show that $\langle x,y \rangle$ is a maximal ideal (and therefore prime). – Ben Grossmann Jul 20 '21 at 00:20