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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?

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@Andr I don't know what you mean by polynomials 'over what'. I just mean polynomials of the form $P = a_0*x^0 + a_1*x^1 + ... + a_n*x^n$, where $a_i$ and $x$ are real. And doesn't the fact they multiply to form a product imply that they are factors? – Matt Munson Feb 4 '13 at 5:47
@AndréNicolas Ok, as I stated in the above commented I am interested exclusively in polynomials over the reals. And I should have stipulated factors other than eg. $P_1(x)P_3(x)$. – Matt Munson Feb 4 '13 at 5:58
Some searchable terms: When K is a field, K[x] is a PID and hence a UFD. – anon Feb 4 '13 at 6:08
@anon I am familiar with fields only very vaguely, and with the other terminology not at all. Right now, I'm mostly interested in finding partial fraction decompositions for doing integrals over real numbers (basic calculus). – Matt Munson Feb 4 '13 at 6:14
@AndréNicolas Ok cool, I see what your saying. Thanks. – Matt Munson Feb 4 '13 at 6:24
up vote 1 down vote accepted

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.

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Awesome. I am recently familiar with the unique factorization theorem, which is what started me thinking about this. Its pretty cool how you get this similarity between polynomials and integers. – Matt Munson Feb 5 '13 at 2:26

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