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I'm writing a program that converts a text input involving addition, subtraction, multiplication, division, and exponentiation of integers and $i$ into a simplest form, but I've come to realize that I don't know what that should be in some cases.

In particular, what is usually done with surds that have complex numbers inside? I was thinking that it would be desirable to get the whole expression into $a + bi$ form, where $a$ and $b$ are real, but if I'm not mistaken that would require using Euler's formula and polar form in some cases, which in general introduce transcendental functions.

Is there a standard "simplest"/canonical form that sticks exclusively to algebraic representations?

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Computing with algebraic numbers in general is possible but quite difficult to implement. Representing the algebraic numbers is part of the problem and is typically done by root isolation: a number $z$ is represented by a polynomial $f$ that has $z$ as a root together with a description of an open set $U$ say (e.g., a circle or a rectangle) such that $z$ is the unique root of $f$ in $U$. Here's a link to one paper on this (found by googling for "computing with algebraic numbers" that compares this approach with another based on approximating sequences.

As far as I know, there is no much better way to handle algebraic numbers in general. In particular, I don't believe any of the known methods give a canonical form (instead they give an algorithm for equality testing). I don't know of any useful general subclass of the algebraic numbers that admits a more straightforward representation. (Obviously you can do much better for very specific number fields such as $\Bbb{Q}[\sqrt{2}]$.)

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  • $\begingroup$ I suppose what I'm dealing with is a strict subset of the algebraic numbers, in that it excludes the roots of polynomials that aren't solvable by radicals. Is there a name for that subset? $\endgroup$ Nov 20, 2017 at 21:29
  • $\begingroup$ I don't know of a name for that subset .You might like to read about radical extensions. $\endgroup$
    – Rob Arthan
    Nov 20, 2017 at 21:57

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