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>> Really? I was sure an algebraic number is any number computable using
>> only addition, subtraction, multiplication, division and extraction of
>> roots.
>
> http://en.wikipedia.org/wiki/Algebraic_number#Numbers_defined_by_radicals
Right. I think I see what happened here.
An algebraic number is any number that is the solution to a polynomial
equation. And since I thought that all such numbers are constructable by
radicals, I assumed "solution of a polynomial" and "constructable by
radicals" were equivilent definitions for an algebraic number.
Apparently they aren't equivilent. (Which still seems deeply weird to me...)
As I say, I thought the problem was that no single formula could
encompass the solutions to every possible 5th-degree polynomial. I
didn't realise that there are single, fixed polynomials who's solutions
are simply inexpressible.
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