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I need to show that the Galois extension (the separable and normal) extension of the polynomial $f(x) \in \Bbb Q[x]$: $f(x)=x^5-10x+5$ its Galois group is isomorphic to $S_5$

How to do it when I can't even find it's roots?

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    $\begingroup$ If it's $S_5$, then you won't be able to find its roots $\endgroup$ – Kenny Lau Jul 5 '18 at 8:11
  • $\begingroup$ You can find the roots, actually, just not with radicals alone. In any event you need to find only the number of real roots (see the answer), which can be pinned down with Descartes' Rule of Signs plus the easily seen sign change in the polynomial between $x=0$ and $x=1$. $\endgroup$ – Oscar Lanzi Jul 8 '18 at 15:15
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It has exactly $3$ real roots, and $5$ is prime, so it follows.

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