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

what's the rationality of the roots of polynomial related to the rationality of its coefficient? Here is the question just came up to me, is that true for a polynomial with irrational coefficient cannot has a rational root. Is any theorem about this?

share|cite|improve this question
Hint $\rm\ \pi x\ $ has a rational root. – Math Gems Mar 3 '13 at 18:14

As another example, the polynomial $$ x^2 - (\pi +1) x + \pi $$ has $1$ as a root, and no multiple of it by a nonzero constant has all coefficients rational.

If you want all coefficients to be non-rational, (see comment by @MartinBrandenburg) $$ \pi x^2 - (\pi^2 +\pi) x + \pi^2 $$ will do.

share|cite|improve this answer
This has also a rational coefficient. – Martin Brandenburg Mar 3 '13 at 17:04
@MartinBrandenburg, ok, thanks, but multyplying by a constant of course you can make any polynomial monic. – Andreas Caranti Mar 3 '13 at 17:06
@AndreasCaranti but if you put it into an equation, its the same as the previous one – Dylan Zhu Mar 3 '13 at 17:10
@AndreasCaranti anyway you have already give the answer what i want, thanks – Dylan Zhu Mar 3 '13 at 17:11
@DylanZhu, you're welcome! – Andreas Caranti Mar 3 '13 at 17:12

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.