The maximal ideals of are all ideals of the form where is an irreducible polynomial in .
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:
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Determine all the maximal ideals of the ring K[X],
where K is a field and X is an indeterminate.
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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