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Ideal Structure of Rings of Analytic Functions with non-Archimedean Metrics
Author Info
Bruno, Nicholas
Permalink:
http://rave.ohiolink.edu/etdc/view?acc_num=osu1620687935873852
Abstract Details
Year and Degree
2021, Doctor of Philosophy, Ohio State University, Mathematics.
Abstract
O. Helmer’s work applied algebraic methods to the field of complex analysis when he proved the ring of entire functions in the complex plane is a Bezout Domain (i.e., all finitely generated ideals are principal). This inspired M. Henriksen’s work which proved a correspondence between the maximal ideals within the ring of entire functions and ultrafilters on sets of zeros as well as a correspondence between the prime ideals and growth rates on the multiplicities of zeros. We prove analogous results in rings of analytic functions in the non-Archimedean context: all finitely generated ideals in the ring of analytic functions on a disk (or an annulus) of a characteristic zero non-Archimedean field are two-generated but not guaranteed to be principal. We also classify the prime, maximal, closed, and divisorial ideals. We prove the maximal and prime ideal structure in the non-Archimedean context is similar to that of the ordinary complex numbers; however, the methodology has to be significantly altered to account for the failure of Weierstrass factorization on balls of finite radius in fields which are not spherically complete, which was proven by Lazard.
Committee
Kenneth Alan Loper, PhD (Advisor)
Ivo Herzog, PhD (Committee Member)
Cosmin Roman, PhD (Committee Member)
Steven MacEachern, PhD (Other)
Subject Headings
Mathematics
Keywords
p-adic
;
ideals
;
non-Archimedean
;
Bezout
;
analytic functions
;
zero sets
;
divisors
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Refworks
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RIS
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Citations
Bruno, N. (2021).
Ideal Structure of Rings of Analytic Functions with non-Archimedean Metrics
[Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1620687935873852
APA Style (7th edition)
Bruno, Nicholas.
Ideal Structure of Rings of Analytic Functions with non-Archimedean Metrics.
2021. Ohio State University, Doctoral dissertation.
OhioLINK Electronic Theses and Dissertations Center
, http://rave.ohiolink.edu/etdc/view?acc_num=osu1620687935873852.
MLA Style (8th edition)
Bruno, Nicholas. "Ideal Structure of Rings of Analytic Functions with non-Archimedean Metrics." Doctoral dissertation, Ohio State University, 2021. http://rave.ohiolink.edu/etdc/view?acc_num=osu1620687935873852
Chicago Manual of Style (17th edition)
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Document number:
osu1620687935873852
Download Count:
105
Copyright Info
© 2021, some rights reserved.
Ideal Structure of Rings of Analytic Functions with non-Archimedean Metrics by Nicholas Bruno is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Based on a work at etd.ohiolink.edu.
This open access ETD is published by The Ohio State University and OhioLINK.