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Math Help - Maximal ideals of K[X]

  1. #1
    Super Member Bernhard's Avatar
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    Maximal ideals of K[X]

    I am reading R.Y.Sharp's book: "Steps in Commutative Algebra.

    In Chapter 3: Prime Ideals and Maximal Ideals, Exercise 3.6 reads as follows:

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

    Determine all the maximal ideals of the ring K[X],

    where K is a field and X is an indeterminate.

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

    Can someone please help me get started on this problem.

    I suspect we may be able to use the following theorem/lemma:

    "Let I be an ideal of the commutative ring R.

    Then I is maximal if and only if R/I is a field."

    However I am not sure of exactly how to go about utilising this result.

    I would very much appreciate some help.

    Peter
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  2. #2
    Junior Member Nehushtan's Avatar
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    Re: Maximal ideals of K[X]

    The maximal ideals of K[X] are all ideals of the form \langle f(x)\rangle where f(x) is an irreducible polynomial in K[X].
    Thanks from Bernhard
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