Noetherian Rings - Dummit and Foote - Chapter 15 - exercise 9

[Peter]

I am reading Dummit and Foote Chapter 15, Section 15.1: Noetherian Rings and Affine Algebraic Sets.

Exercise 9 reads as follows:

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For k a field show that any subring of a polynomial ring k[x] containing k is Noetherian.

Give an example to show that such subrings need not be UFDs.

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Can someone please help me get started on this problem?

Peter

[Note: This has also been posted on MHF]