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Spring 2016 - Problem 6

Galois theory

Let $K$ be a field of characteristic $p > 0$ . For an element $a \in K$ , show that the polynomial $P\p{X} = X^p - X + a$ is irreducible over $K$ if and only if it has no root in $K$ . Show also that, if $P$ is irreducible, then any root of it generates a cyclic extension of $K$ of degree $p$ .

Solution.

Notice that over the splitting field of $P$ , if $\alpha$ is a root of $P$ , then for $i \in \Z/p\Z$ , we have

\begin{aligned} P\p{\alpha + i} &= \p{\alpha + i}^p - \p{\alpha + i} + a \\ &= \alpha^p + i^p - \alpha - i + a \\ &= \alpha^p - \alpha + a \\ &= P\p{\alpha} \\ &= 0. \end{aligned}

For the third equality, we used the fact that $\p{\Z/p\Z}^\times \simeq \Z/\p{p-1}\Z$ , i.e., $i^p = i$ . Thus, if $P$ has a root in $K$ , then it automatically has all its roots $\alpha, \alpha + 1, \ldots, \alpha + \p{p-1}$ in $K$ as well, so $P$ is irreducible if and only if $P$ has no roots in $K$ .

For the second claim, note that the calculation above shows that the splitting field of $P$ is $K\p{\alpha}$ where $\alpha$ is any root of $P$ . Moreover, it also shows $P$ is separable, so $K\p{\alpha}/K$ is Galois. Hence, $\abs{\Gal\p{K\p{\alpha}/K}} = \br{K\p{\alpha} : K} = p$ , so $\Gal\p{K\p{\alpha}/K} \simeq C_p$ , which was what we wanted to show.