Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I know of a theorem in algebra, that every polynomial $p\left(X\right)=a_{0}+a_{1}X+\ldots+a_{n}X^{n}$ that is nonconstant and has real coefficients, admits a factorisation of the form $$ p\left(X\right)=c\left(X-\lambda_{1}\right)\ldots\left(X-\lambda_{m}\right)\left(X^{2}+\alpha_{1}X+\beta_{1}\right)\ldots\left(X^{2}+\alpha_{M}X+\beta_{M}\right), $$ where $m+M\geqslant1$, $c,\lambda_{1},\ldots,\lambda_{m}\in\mathbb{R}$ and $\left(\alpha_{j},\beta_{j}\right)\in\mathbb{R}^{2}$ with $\alpha_{j}<4\beta_{j}$ for $j\in\left\{ 1,\ldots,M\right\} $. My questions are:

1) Is there a name for this theorem ?

2) a) Is there an analogue of this theorem for " polynomials consisting of infinite sums", i.e. for Laurent polynomials ? b) Is there an analogue of this theorem for convergent sums ? (So that they be expressed as a convergent product)

share|cite|improve this question
up vote 2 down vote accepted

1) It's a special case of the Fundamental Theorem of Algebra.

2) I think the closest analogue is the Weierstrass Factorization Theorem

share|cite|improve this answer
And to 2)a) - do you know if there is analogue for formal power series? – el le Jun 28 '12 at 18:28

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.