0

Given $p$ is a prime, $k$ is an algebraically closed field of characteristic $p$. and $F = k(t)$, where $t$ is a variable, let $L$ be the splitting field of $x^p − x + t$ over $F$. Then it can be shown that $L$ is not solvable over $F$. Does anyone know where to find a proof or how to prove. I cannot seem to find the proof even though it seems like a standard result in Galois Theory~

Homaniac
  • 1,215
  • 4
    The splitting field is cyclic, if $a$ is a root then so is $a+1$, so $a\mapsto a+1$ generates the Galois group (cyclic of order $p$). Look up Artin-Schreier extensions for more info. – user8268 Nov 09 '18 at 06:51
  • The polynomial has a zero $x=t+t^p+t^{p^2}+t^{p^3}+\cdots$ in the ring of formal power series $k[[t]]$. You get the other zeros by Artin-Schreier as described by user8268. All cyclic degree $p$ extensions are actually Artin-Schreier (in characteristic $p$). So the splitting field is surely solvable. I don't think it's a root tower extension though. May be that's what you are asking? – Jyrki Lahtonen Nov 10 '18 at 05:20

0 Answers0