# Mock covers and Galois theory over complete domains

#### Abstract

Consider an integral domain R that is complete with respect to a non-zero prime ideal a. This thesis proves two Galois-theoretic results about such domains. Using Grothendieck's Existence Theorem we prove that every finite group occurs as the Galois group of a cover of the line over cx R, i.e. a cover of $\IP\sbsp{\ R}{1}.$ The cover corresponds to a Galois extension of Frac(R (x)), thus answering the Inverse Galois Problem affirmatively for these domains. This work generalizes results of David Harbater who proved the result in the case where the ideal is maximal and the domain is normal. As a consequence, we deduce that if R is a power series ring of dimension at least two with coefficients in a domain, then every finite group occurs as a Galois group over Frac(R). This answers the IGP affirmatively for such power series domains and proves a conjecture posed by Moshe Jarden.

#### Subject Area

Mathematics

#### Recommended Citation

Lefcourt, Tamara R, "Mock covers and Galois theory over complete domains" (1996). *Dissertations available from ProQuest*. AAI9627955.

https://repository.upenn.edu/dissertations/AAI9627955