2026-09-23T04:02:15Z https://www.ideals.illinois.edu/oai-pmh

oai:www.ideals.illinois.edu:2142/127304 2025-03-29 col_2142_5131 col_2142_16340 com_2142_5130 com_2142_16339 com_2142_8903

application/pdf en text Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2025-03-28 without embargo terms The student, Malachi Robinson, accepted the attached license on 2024-12-13 at 11:37. The student, Malachi Robinson, submitted this Thesis for approval on 2024-12-13 at 11:43. This Thesis was approved for publication on 2024-12-13 at 14:46. DSpace SAF Submission Ingestion Package generated from Vireo submission #21597 on 2025-03-28 at 14:28:54 Sturm sequences of polynomials related to regular primes Robinson, Malachi M. 2024-12-13 Ahlgren, Scott Sturm Sequence eng We discuss Kummer’s algebraic definition of a regular prime number p involving the class number of the cyclotomic field Q(ζp) and Kummer’s criterion for a prime number to be regular, which involves the numerators of Bernoulli numbers. A discussion of Sturm sequences and a software implementation follows. We conjecture and provide strong evidence for an integer upper and lower bound, [k − 4 k 5, k − 4 k 6] for the number of real roots of the kth Bernoulli polynomial using data generated by computing Sturm sequences. We then prove explicit formulae for each of the terms of the Sturm sequences of cyclotomic polynomials Φn(x), where n is a prime greater than 3 or a power of 2. 2024-12 Thesis https://hdl.handle.net/2142/127304 Copyright 2024 Malachi Robinson Mathematics Mathematics University of Illinois at Urbana-Champaign M.S. Thesis