Sturm sequences of polynomials related to regular primes
Robinson, Malachi M.
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https://hdl.handle.net/2142/127304
Description
Title
Sturm sequences of polynomials related to regular primes
Author(s)
Robinson, Malachi M.
Issue Date
2024-12-13
Director of Research (if dissertation) or Advisor (if thesis)
Ahlgren, Scott
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
M.S.
Degree Level
Thesis
Keyword(s)
Sturm sequence
Abstract
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.
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