Date of Award
Doctor of Philosophy (PhD)
Dr. James Coykendall
Dr. Matthew Macauley
Dr. Keri Ann Sather-Wagstaff
Dr. Hui Xue
Dr. Michael Burr
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.
Swanson, Joseph, "Algebraic and Integral Closure of a Polynomial Ring in its Power Series Ring" (2023). All Dissertations. 3377.