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I know that the real algebraic numbers $\mathbb A \subset \mathbb R$ form a field. I've seen this as a more theoretical result, but it's also seems nice idea to implement algebraic numbers for the computer in order to make computations more exact.

Now I wonder how could this be done? I thought of representing an algebraic number $\alpha$ as a tuple $(P,(a,b)) \in \mathbb Q[T]\times \mathbb Q^2$ such that $\alpha$ is the unique root of $P$ in $[a,b]$.

Now take for example the numbers

  • $\varphi = (T^2-T-1,(\frac 3 2,2))$
  • $\sqrt 2 = (T^2-2,(1,2))$

But how can I actually perform computations on this numbers?

Surely $\varphi + \sqrt 2$ lies in $(\frac 5 2, 4)$, but what polynomial's root is it? What about $\varphi \cdot \sqrt 2$? In general, given polynomials $P,Q \in \mathbb Q[T]$ with $P(x)=Q(y)=0$, how do I find some polynomial $S$ such that $S(x+y)=0$?

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    I don't remember the method (or even whether or not have I seen a concrete one), but I'm pretty sure that the degree of the polynomial will grow exponentially as you perform operations on algebraic numbers represented that way, so I don't think it would be a good representation to be used by a computer...2012-08-13
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    This is a large and complicated subject. If you want to do a small number of easy computations, Mathematica and Sage (and probably all other computer algebra systems, but those are the ones I know) have routines to do this for you. If you want to understand the underlying theory, see Sasha's answer. If you want to do large examples, or write a library for doing this sort of computation, see Henri Cohen's book "A Course In Computational Algebraic Number Theory".2012-08-13
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    Some useful keywords here are "Kronecker sum" and "Kronecker product."2012-08-13
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    The problem of finding a polynomial vanishing at the sum, or the product, of two algebraic numbers got some very nice answers two months ago at http://math.stackexchange.com/questions/155122/how-to-prove-that-the-sum-and-product-of-two-algebraic-numbers-is-algebraic2012-08-14

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