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

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In summary, a prime ideal in K[X] is a subset of the polynomial ring K[X] that is closed under addition and multiplication by elements in K[X] and has the additional condition that the product of two polynomials in the ideal must also be in the ideal. This sets it apart from a regular ideal, which only needs to be closed under addition and multiplication. Prime ideals can be generated by more than one polynomial and are often used in polynomial factorization by identifying the prime ideals that correspond to the irreducible factors of a given polynomial. However, prime ideals are not unique in K[X] as there can be multiple prime ideals that correspond to the same irreducible polynomial.
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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 $f(X)$?
 

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

1. What is a prime ideal in K[X]?

A prime ideal in K[X] is a subset of the polynomial ring K[X] that satisfies two conditions: it is an ideal (meaning it is closed under addition and multiplication by elements in K[X]), and if the product of two polynomials belongs to the ideal, then at least one of the polynomials must also belong to the ideal.

2. How is a prime ideal different from a regular ideal?

A prime ideal is a special type of ideal that has an additional condition that the product of two polynomials in the ideal must also be in the ideal. This is not a requirement for a regular ideal, which only needs to be closed under addition and multiplication by elements in K[X].

3. Can a prime ideal be generated by more than one polynomial?

Yes, a prime ideal can be generated by more than one polynomial. In fact, prime ideals in K[X] are often generated by multiple polynomials. For example, the ideal (x, x^2) in K[x] is prime and is generated by two polynomials.

4. How can prime ideals be used in polynomial factorization?

Prime ideals can be used in polynomial factorization by identifying the prime ideals in K[X] that correspond to the irreducible factors of a given polynomial. By factoring a polynomial into its irreducible components, we can use the prime ideals associated with each factor to reconstruct the original polynomial.

5. Are prime ideals unique in K[X]?

No, prime ideals are not unique in K[X]. There can be multiple prime ideals that correspond to the same irreducible polynomial. For example, in K[x], both (x) and (x^2) are prime ideals that correspond to the irreducible polynomial x.

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