Why is (x) a prime ideal in k[x,y]?

[Math Amateur]

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

 

[R136a1]

Use the first isomorphism theorem to show that $k[X,Y]/(X)$ is an integral domain. The right function is the evaluation in $0$:

$$k[X,Y]\rightarrow k[Y]:P(X,Y)\rightarrow P(0,Y)$$

 

[Math Amateur]

Thanks for the help, r136a1

Just checking and reflecting on the use of the First Isomorphism Theorem

Peter