I am looking for examples where the roots of a 5 degree polynomial in $Q[x]$ cannot be expressed in terms of radicals (that part is easy to acchieve, there are a lot of examples) but they CAN be expressed in terms of something else. In other words, I am looking for an example where I can see the roots in an explicit way and where we can explicitly see that they are not in terms of radicals.
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I believe you mean 5th degree integer/rational polynomial (definitely not real polynomial) – Wen Dec 15 '17 at 01:31
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"Radicals" as in square roots only, or $n$th roots? – puzzleweaver Dec 15 '17 at 01:31
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1you might like these http://math.stackexchange.com/questions/1996552/any-more-cyclic-quintics – Will Jagy Dec 15 '17 at 01:33
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1Here's a paper you may find interesting: https://arxiv.org/pdf/1308.0955.pdf – nkm Dec 15 '17 at 01:42
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Well, since the OP never edited...
$x^5-\pi$ is a perfectly valid example as $\sqrt[5]\pi$ is a root which is transcendental
Wen
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