#### Publication Date

7-12-2014

#### Abstract

A remarkable and special Galois Theory appears from the study of the arithmetic analogue of ordinary differential equations; where functions are replaced by integers, the derivative operator replaced by the Fermat quotient operator' and differential equations are replaced by arithmetic differential equations. The main result presented in the thesis will be the study of a very special class of arithmetic subgroups of GL_n. We also introduce a set of functions, that we call Leibniz systems. These functions 'generate' some examples of the differential subgroups of GL_n. As a by-product we found more analogies between the ordinary differential operator and the Fermat quotient operator, such as the chain rule and the product rule.

#### Degree Name

Mathematics

#### Level of Degree

Doctoral

#### Department Name

Mathematics & Statistics

#### First Advisor

Buium, Alexandr

#### First Committee Member (Chair)

Boyer, Charles

#### Second Committee Member

Vassilev, Janet

#### Third Committee Member

Miller, Lance

#### Language

English

#### Keywords

Arihmetic differential Subgroups og GL_{n}

#### Document Type

Dissertation

#### Recommended Citation

Heras-Llanos, Alfonso E.. "Arithmetic Differential Subgroups of GL_{n}." (2014). http://digitalrepository.unm.edu/math_etds/19