I am wondering how to show that the ideal (x,y) is prime and maximal in $\mathbb{Q}[x,y]$.
Prime Ideal of a Polynomial Ring
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ring-theory
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7Look at the ring homomorphism $\varphi: \mathbb{Q}[x,y] \rightarrow \mathbb{Q}$ such that $\varphi(f)=f(0,0)$. What is the kernel of this map? – 2011-01-27
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2@Prometheus: the kernel is exactly (x,y) and the image of the homomorphism is Q, and then using the first isomorphism to conclude that (x,y) is maximal because Q[x,y]/(x,y) is a field – 2011-01-27