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Prime ideal (x) in k[x,y]

  1. Nov 12, 2013 #1
    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
     
  2. jcsd
  3. Nov 12, 2013 #2
    Use the first isomorphism theorem to show that ##k[X,Y]/(X)## is an integral domain. The right function is the evaluation in ##0##:

    [tex]k[X,Y]\rightarrow k[Y]:P(X,Y)\rightarrow P(0,Y)[/tex]
     
  4. Nov 12, 2013 #3
    Thanks for the help, r136a1

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

    Peter
     
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