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

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

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.