# Math Help - Prime ideal (x) in k[x,y]

1. ## Prime ideal (x) in k[x,y]

Example (2) on page 682 of Dummit and Foote reads as follows:

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(2) For any field k, the ideal (x) in k[x,y] is primary since it is a prime ideal.

... ... etc

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Now if (x) is prime then obviously (x) is primary BUT ....

How do we show that (x) is prime in k[x, y]?

Would appreciate some help.

Peter

2. ## Re: Prime ideal (x) in k[x,y]

Here's a quick way of doing so:

$K[X,Y]/(X) \cong K[Y]$ and as $K[Y]$ is an integral domain, we have that (X) is a prime ideal