TITLE: How to find a polynomial from a given root?
QUESTION [8 upvotes]: I was asked to find a polynomial with integer coefficients from a given root/solution.
Lets say for example that the root is: $\sqrt{5} + \sqrt{7}$.  

How do I go about finding a polynomial that has this number as a root?
Is there a specific way of finding a polynomial with integer coefficients?

Any help would be appreciated. Thanks.

REPLY [4 votes]: Let me just remark that there are algorithms for doing this even when the number is not given in terms of radicals but only as a decimal expansion (but is suspected to be algebraic). Namely, one uses the LLL algorithm to try to construct an integral linear combination of $1,\alpha,\alpha^2,\alpha^3,\ldots$ that is zero up to a rounding error. If no such linear combination is found then one can conclude that the number is either transcendental, or the minimal polynomial must have enormous coefficients or enormous degree.
See http://en.wikipedia.org/wiki/LLL_algorithm#Applications