What can we say about roots of a quintic that is not solvable by radicals?

Asked May 05 '17 at 00:20

Active May 05 '17 at 00:20

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If a polynomial $f(x) \in \mathbb{Q}[x]$ is not solvable by radicals, then what can we say about the roots of this polynomial? How might we get our hands on them? Is there anything known?

The roots are algebraic numbers (obviously) and they live in some finite extension of $\mathbb{Q}$.

asked May 05 '17 at 00:20

User0112358

What properties of the roots are you interested in? We can still say they satisfy Vieta's formulas, for example. We can also talk about the Galois group of the field extension they generate. – Tob Ernack May 05 '17 at 00:23
They can be expressed as elliptic or hypergeometric functions of the coefficients, maybe? – Oscar Lanzi May 05 '17 at 00:26
https://math.stackexchange.com/questions/540964/how-to-solve-fifth-degree-equations-by-elliptic-functions – Kaj Hansen May 05 '17 at 00:33
@OscarLanzi really? That is the something I am looking for. Where can I read more about that fact? – User0112358 May 05 '17 at 00:34
1

@KajHansen thanks :) – User0112358 May 05 '17 at 00:35

What can we say about roots of a quintic that is not solvable by radicals?

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