MHB Prime Ideals in K[X] - Commutative Algebra Exercise 3.22 (ii)

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The discussion centers on identifying all prime ideals in the ring K[X], where K is a field and X is an indeterminate. It is noted that K[X] is a Principal Ideal Domain (PID), meaning every ideal is generated by a single element. For an ideal P = (f(X)) to be prime, the polynomial f(X) must be irreducible over the field K. The conversation seeks clarification on the implications of f(X) being prime and how to approach the exercise effectively. Understanding these properties is essential for solving the problem.
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I am reading R.Y.Sharp's book: "Steps in Commutative Algebra.

In Chapter 3: Prime Ideals and Maximal Ideals, Exercise 3.22 (ii) reads as follows:

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Determine all the prime 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.

Peter
 
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$K[X]$ is a PID. So any ideal is principal.

If $P = (f(X))$ is prime, what can you say about $f(X)$?
 
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