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