Let $\alpha, \beta \in \mathbb{F}_{q}$, show that $x^q -\alpha x - \beta$ is irreducible $\iff$ $\beta \ne 0, \alpha = 1$, and $q$ is prime. I am having trouble proving the forward direction. For the reverse direction, I know that $x^q-x-\beta$ has no roots in $\mathbb{F}_{q}[x]$ but I am not sure how this implies irreducibility.
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Please try to make the titles of your questions more informative. For example, Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. You can find more tips for choosing a good title here. – Shaun Oct 01 '19 at 23:41
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That's a duplicate of https://math.stackexchange.com/questions/3378637/prove-implications-of-irreducible-polynomial – principal-ideal-domain Oct 04 '19 at 05:07