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# Homology, Homotopy and Applications

## Volume 10 (2008)

### Number 3

### Proceedings of a Conference in Honor of Douglas C. Ravenel and W. Stephen Wilson

### Galois extensions of Lubin-Tate spectra

Pages: 27 – 43

DOI: https://dx.doi.org/10.4310/HHA.2008.v10.n3.a3

#### Authors

#### Abstract

Let $E_n$ be the $n$-th Lubin-Tate spectrum at a prime $p$. There is a commutative $S$-algebra $E^{\mathrm{nr}}_n$ whose coefficients are 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 $E^{\mathrm{nr}}_n$ 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 are no non-trivial connected Galois extensions of $E^{\mathrm{nr}}_n$ with Galois group a finite group $G$ with cyclic quotient. Our results carry over to the $K(n)$-local context.

#### Keywords

Galois extensions, separable closure, Witt vectors, Lubin-Tate spectra

#### 2010 Mathematics Subject Classification

13B05, 55N22, 55P43, 55P60

Published 1 January 2008