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

  1. #1
    Super Member Bernhard's Avatar
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    Prime ideal (x) in k[x,y]

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

    ------------------------------------------------------------------------

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

    ... ... etc

    ----------------------------------------------------------------------------

    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
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  2. #2
    Member Haven's Avatar
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    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
    Thanks from Bernhard
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