Unique factorization in polynomial rings

2010-02-12T20:40:04Z

Everybody knows that polynomial rings over fields have unique factorization, and that if $R$ has unique factorization, then so does $R[X]$. And everybody knows who proved these results first.

Well, perhaps not. Which brings me to my questions:

Who first proved that $k[x]$ is factorial if $k$ is a field?
Who first proved that $R[X]$ is factorial if $R$ is?

I'm not really interested in the correct answers (which I think I know) but in what people believe are the right answers, so I made this community wiki. In particular I'd like to read what the non-specialists think, even if it only concerns a special case such as $R = \mathbb Z$. Of course I promise to give my own answer in a couple of days. And kudos to everyone who turns out to know more about this than I do -)

Answer by Franz Lemmermeyer for Unique factorization in polynomial rings

2010-02-13T05:47:01Z

This doesn't fly. If the powers that be would like to delete the question, please go ahead. For my defense I'd like to add that I wanted to see whether my guess that most people would use the name Gauss in their answer holds water or not. So here's what I think I know.

Polynomials became mathematical objects through the work of the Italians (Cardano etc.); after preliminary work done by Pedro Nunez, Simon Stevin showed that there is a Euclidean algorithm in polynomial rings that allows one to compute greatest common divisors. Strictly speaking it is difficult to separate unique factorization in ${\mathbb R}[X]$ or ${\mathbb C}[X]$ from the fundamental theorem of algebra, but certainly those who were working on the latter (d'Alembert, Euler) did not mention unique factorization anywhere.

The concept of unique factorization is due to Gauss (1801), although partial credit should be given to Euclid. Gauss proved that the rings $\mathbb Z$ and ${\mathbb Z}[i]$ are factorial, and did the same for ${\mathbb F}_p[X]$ in his famous Section VIII of the Disquisitiones, which was published posthumously. Dirichlet realized in the 1840s that Euclidean domains are factorial and stated this as explicit as he could. But noone seemed to put 1 and 1 together to derive (1); my guess is that its essential content was known to people like Dirichlet, Eisenstein, Dedekind and Kronecker, but the result does not appear anywhere except much later when Weber wrote his textbook on algebra. Let me also add that Dirichlet could state that Euclidean rings are factorial even though the concept of an abstract ring came much later (he said something to the effect that if there is a Euclidean division algorithm, then you must have unique factorization no matter which "domain" you are working in).

Kronecker, in his lectures and, somewhat later, also in his publications, proved that the polynomial rings with finitely many variables and coefficients from $\mathbb Z$ have unique factorization. The first explicit statement (and proof) of (2) that I know is in Hensel's article Über eindeutige Zerlegung in Primelemente, J. Reine Angew. Math. 158 (1927), 195--198. Again I guess that this wasn't exactly news for Emmy Noether or Artin, and the result is mentioned in just about every textbook on algebra, starting with van der Waerden's algebra published in 1930, which was based on lectures by Noether and Artin during the 1920s.

Corrections are welcome.

Unique factorization in polynomial rings - MathOverflow

Unique factorization in polynomial rings

Answer by Franz Lemmermeyer for Unique factorization in polynomial rings