I want to get a quintic polynomial $f(X) \in \mathbb{Q}[X]$ whose Galois group $\mathrm{Gal}(L/\mathbb{Q}) \cong S_5$ where $L$ is the splitting field of $f(X)$.

One of strategies to get it is finding an irreducible polynomial $f(X) \in \mathbb{Q}[X]$ which has exactly three real roots. But, I can't do it.

Can you give me examples of quintic polynomial $f(X) \in \mathbb{Q}[X]$ which has exactly three real roots and is irreducible over $\mathbb{Q}$ $??$

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    $\begingroup$ A lot of textbooks on Galois theory have them. $\endgroup$ – Angina Seng Apr 15 '19 at 6:50

You can take $p(x)=x^5-4x-2$. It is irreducible in $\mathbb Q[x]$, by Eisenstein's criterion. And it is easy to deduce from the fact that $p'(x)=5x^4-4$ and from the intermediate value theorem that it has $3$ and only $3$ real roots.

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  • $\begingroup$ I understand your answer. Thanks for your help. $\endgroup$ – 神宮寺春姫 Apr 15 '19 at 7:01

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