# Maximal ideals of K[X]

• Dec 11th 2013, 02:13 PM
Bernhard
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:

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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
• Dec 29th 2013, 11:12 AM
Nehushtan
Re: Maximal ideals of K[X]
The maximal ideals of $\displaystyle K[X]$ are all ideals of the form $\displaystyle \langle f(x)\rangle$ where $\displaystyle f(x)$ is an irreducible polynomial in $\displaystyle K[X]$.