Polynomials in n indeterminates and UFDs

[Math Amateur]

In the introduction to Chapter 1 of his book "Introduction to Plane Algebraic Curves", Ernst Kunz states that the polynomial ring $K[ X_1, X_2, \ ... \ ... \ , X_n]$ over a field $K$ is a unique factorization domain ... ... but he does not prove this fact ...

Can someone demonstrate a proof of this proposition ... or point me to a text or online notes that contain a proof ...

Help will be appreciated ... ...

Peter

 

[fresh_42]

You can show it for one variable and proceed by induction, because $K[X_1,X_2, \dots , X_n] = K[X_1,X_2, \dots , X_{n-1}][X_n].$
For $K[X]$ you know that you can apply the Euclidean algorithm (division) to find all irreducible factors of a polynomial.

 

[Math Amateur]

You can show it for one variable and proceed by induction, because $K[X_1,X_2, \dots , X_n] = K[X_1,X_2, \dots , X_{n-1}][X_n].$
For $K[X]$ you know that you can apply the Euclidean algorithm (division) to find all irreducible factors of a polynomial.

Hmm ... yes, get the general idea ... but not quite sure how the induction is set up and how exactly it proceeds ... Thinking ...

Thanks for the help ...

Peter

 

[fresh_42]

Hmm ... yes, get the general idea ... but not quite sure how the induction is set up and how exactly it proceeds ... Thinking ...

Thanks for the help ...

Peter

The induction step is: If a ring $R$ is UFD, so is $R[X]$.
(See http://math.harvard.edu/~waffle/ufds2.pdf ) This brief article also contains a pretty good overview on some frequent classes of rings. I think you should read it to gain a feeling for the concepts and a pool of examples (14 pages).

 

[Math Amateur]

The induction step is: If a ring $R$ is UFD, so is $R[X]$.
(See http://math.harvard.edu/~waffle/ufds2.pdf ) This brief article also contains a pretty good overview on some frequent classes of rings. I think you should read it to gain a feeling for the concepts and a pool of examples (14 pages).

Thanks fresh_42 ... ... most helpful ... appreciate the help ...

Peter

 

[mathwonk]

auslander and buchsbaum were among the foremost experts of unique factorization domains. their graduate algebra book, is available free here ( see e.g. chapter 5):

https://babel.hathitrust.org/cgi/pt?id=mdp.39015015615886;view=1up;seq=10my own discussion is in sections 4,5 of these free class notes.

http://alpha.math.uga.edu/%7Eroy/844-1.pdfand by the way Buchsbaum is apparently still active:

http://people.brandeis.edu/~buchsbau/