Galois extensions of Lubin-Tate spectra

Abstract

Let $E_n$ be the $n$-th Lubin-Tate spectrum at a prime $p$ . There is a commutative $S$-algebra $E^{\rm{nr}}_n$ whose coefficients a.re built from the coefficients of $E_n$ and contain all roots of unity whose order is not divisible by $p$. For odd primes $p$ we show that $E^{\rm{nr}}_n$ does not have any non-trivial connected finite Galois extensions and is thus separably closed in the sense of Rognes. At the prime 2 we prove that there a.re no non-trivial connected Galois extensions of $E^{\rm{nr}}_n$ with Galois group a finite group $G$ with cyclic quotient. Our results carry over to the $K(n)$-local context.