# Unique factorization in polynomial rings

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:

1. Who first proved that $k[x]$ is factorial if $k$ is a field?
2. 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 -)

• The immediately tempting answers are 1. Euclid and 2. Gauss. I expect that 1. is probably too early (while correct in spirit, I don't think the ancient Greeks worked with polynomials) and 2. is probably too late (since the result is a corollary of something called Gauss' Lemma, if the correct answer were Gauss, the question would not be so interesting). I am hoping that I may be closer to correct on average Feb 12 '10 at 21:24
• Read literally, one cannot even state (let alone prove) #1 without having a notion of a "field" which I imagine would disqualify both Euclid and Guass. The first general definition of a field is by Weber (1891) according to Wikipedia. Earlier notions were things like a subfield of the complex numbers. I'm not sure if the question assumes that the prover knew they were working over general fields, or rather looking for proofs which are essentially independent of the base field (even if they were formulated over a specific field). Feb 12 '10 at 21:52
• You're not interested in the correct answers, but in the opinions of non-specialists about factual information. What are you trying to accomplish? I can only imagine somebody answering this question if they were looking to be controversial, since you're pre-emptively implying in your question that they are wrong. I would vote to close this as not a real question. Feb 13 '10 at 0:53
• I hadn't really thought about this before, but my guesses are: 1) Viete -- since he seemed to be the first to formulate things in a way that we might recognize as algebra. 2) Even though everyone (or it should be everyone) knows about Gauss's lemma which is the key part in the modern proof, if I had to guess I might say Euler. Feb 13 '10 at 1:47

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.

• Franz- We definitely won't delete the question, and I wouldn't even vote to close it, but I hope you understand why many of us thought it was a strange question to ask. It sure seems like what you wanted to do was tell us some cool history of mathematics, which is a fine thing to do in general, but not the purpose of MathOverflow. That's great stuff to put on a blog or website, but it doesn't really make it a question. Feb 13 '10 at 7:19
• Nice answer! My first guess was Noether for (2), but it is good to know the history of this. Feb 13 '10 at 7:25
• Perhaps my comment was too terse. In my defense, I wasn't so much trying to get the question closed as to explain why I don't care for it and to get you to explain it better to me. The main problem is that I don't understand the goal; it looks like either an attempt to start a math history discussion or some sort of social experiment, neither of which make sense on MO. The other problem is that it reads like you're handing out a quiz for a class nobody realized they were enrolled in, asking people to take the quiz especially if they are unlikely to know the right answers; it's patronizing. Feb 13 '10 at 19:05
• No offense taken. Feb 14 '10 at 12:25