Can the product of irreducible polynomials have non-constant factors other than those polynomials?

Question

Can the product of irreducible polynomials over the reals, $P_1, P_2,...,P_n$, have non-constant polynomial factors other than those polynomials or products of them (eg. $P_1P_3$)? It seems that the answer should be no, and we know that that holds for integers, but where is the proof?

Answer 1

Are you familiar with the Unique Factorization Theorem for the integers? The one that says that every positive integer can be written as a product of primes, and that the product is unique, up to the order in which you write down the primes?

Well, there is an analogous theorem for polynomials. Every monic polynomial with coefficients in the reals (or, indeed, in any field, $K$) can be written as a product of monic irreducibles, and the product is unique, up to the order in which you write down the irreducibles.

There is a proof very much like the one usually used for the integers. You start by proving the Division Theorem, which says you can divide any polynomial by any non-zero polynomial and get a remainder that's either zero or of degree less than the divisor. Then repeated applications of the Division Theorem establish the Euclidean Algorithm for finding the greatest common divisor of any two polynomials (as long as they are not both identically zero). You use that to show that if an irreducible polynomial divides a product of two polynomials then it must divide at least one of the two polynomials. From there it's an easy step to the Unique Factorization Theorem. You'll find it in any text that does fields, and you'll find the prime number version in any text that does Number Theory (not to mention any of several thousand websites that you will find if you type "unique factorization" into the web).

And, once you have unique factorization, you can answer your question, just as you would for the integers.