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$F$ is a field of characteristic $p$ and $a\neq c^p-c$ for $c\in F$. Then determine the galois group of $x^p-x-a$.

First I showed that this is an irreducible polynomial and has no multiple roots. This will imply that the polynomial had $p$- distinct roots in its splitting field. Hence we can get a $p$-cycle. This means that the Galois group is $S_p$. Is my reasoning correct?

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    $\begingroup$ Your $c^p - a$ should be $c^p - c.$ Concerning the question, just having a $p$-cycle is hardly sufficient. A separable irreducible cubic has Galois group $A_3$ or $S_3$. Both contain a $3$-cycle. You should think about how the roots are related to each other. If you have one root $\alpha$ then find a formula for the remaining roots in terms of $\alpha$. $\endgroup$ – KCd Apr 26 '15 at 0:46
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Actually, if $\alpha$ is a root, then so is $\alpha+n$, for all $n\in\mathbb F_p$. Thus, adjoining one root gives the entire splitting field, and thus the splitting field is an extension of degree $p$. The Galois Group is thus isomorphic to $C_p$.

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