Galois extensions of Lubin-Tate spectra

TL;DR

This paper investigates the Galois extensions of Lubin-Tate spectra, showing that at odd primes, certain extensions are trivial, and at prime 2, specific Galois groups are excluded, with implications for the K(n)-local setting.

Contribution

It proves the absence of non-trivial connected Galois extensions of Lubin-Tate spectra under specific conditions at odd primes and prime 2.

Findings

No non-trivial connected Galois extensions at odd primes.

No non-trivial connected Galois extensions with cyclic quotient Galois group at prime 2.

Results extend to the K(n)-local context.

Abstract

Let E_n be the n-th Lubin-Tate spectrum at a prime p. There is a commutative S-algebra E^{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^{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^{nr}_n with Galois group a finite group G with cyclic quotient. Our results carry over to the K(n)-local context.