#### Date of Award

8-2023

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematical Sciences

#### Committee Chair/Advisor

Dr. James Coykendall

#### Committee Member

Dr. Matthew Macauley

#### Committee Member

Dr. Keri Ann Sather-Wagstaff

#### Committee Member

Dr. Hui Xue

#### Committee Member

Dr. Michael Burr

#### Abstract

Let R be a domain. We look at the algebraic and integral closure of a polynomial ring, R[x], in its power series ring, R[[x]]. A power series α(x) ∈ R[[x]] is said to be an algebraic power series if there exists F (x, y) ∈ R[x][y] such that F (x, α(x)) = 0, where F (x, y) ̸ = 0. If F (x, y) is monic, then α(x) is said to be an integral power series. We characterize the units of algebraic and integral power series. We show that the only algebraic power series with infinite radii of convergence are

polynomials. We also show which algebraic numbers appear as radii of convergence for algebraic power series.

Additionally, we provide a new characterization of algebraic power series by showing that a convergent power series, α(x), is algebraic over L if and only if α(a) is algebraic over L(a) for every a in the domain of convergence of α(x), where L is a countable subfield of C.

#### Recommended Citation

Swanson, Joseph, "Algebraic and Integral Closure of a Polynomial Ring in its Power Series Ring" (2023). *All Dissertations*. 3377.

https://tigerprints.clemson.edu/all_dissertations/3377