>> 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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