Timeline for When is $f(x^d)$ irreducible?
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| Jun 5, 2015 at 22:37 | comment | added | Lubin | And of course, @KConrad, as an admirer of Lang I should have been aware of this. | |
| Jun 5, 2015 at 16:08 | comment | added | KConrad | There's a section in Lang's Algebra describing exactly when $x^m - a \in F[x]$ for $a \in F^\times$ is irreducible: it is necessary and sufficient that for every prime $\ell$ dividing $m$ we have $a \not\in F^\ell$, and if $4$ divides $m$ we also need $a \not\in -4F^4$. Lang doesn't give a name to this theorem, but it is usually attributed to Vahlen and Capelli (although they did not prove it in the context of an absolutely general field $F$). | |
| Jun 5, 2015 at 15:17 | history | answered | Lubin | CC BY-SA 3.0 |