A note on irregular prime polynomials in cyclotomic function field theory
type: Item
instance of: scholarly article
title: A note on irregular prime polynomials in cyclotomic function field theory
zbMATH Open document ID: 0578.12012
DOI: 10.1016/0022-314X(86)90071-5
published in: Journal of Number Theory
publication date: 1986-01-01T00:00:00Z
review text: Let \(A=\mathbb F_r[T]\), \(r=p^n\), be the polynomial ring over the field with \(r\) elements. Inside \(A\) one has the elements \(\beta (i)\) which are analogs of classical Bernoulli numbers in that they determine the divisibility of class numbers by \(p\). Let \(f\) be a prime of \(A\) of degree \(d\). Then \(f\) is said to be irregular of the first kind if there is some \(i<r^d-1\), \(i\not\equiv 0 (r-1)\), and \(f\mid \beta (i)\); \(f\) is said to be irregular of the second kind if there is a non-zero \(i<r^d-1\), \(i\equiv 0 (r-1)\) and \(f\mid \beta (i)\). The reviewer and M. Rosen had shown that for \(r=p>2\) there are an infinite number of irregular primes of the first kind. In the paper being reviewed the author extends the above result to the case where \(r>2\). Moreover, the author gives a very elegant proof that for arbitrary \(r\), there are an infinite number of irregular primes of the second kind.
reviewed by: David Goss
Mathematics Subject Classification ID: 11R60
Mathematics Subject Classification ID: 11R29
zbMATH DE Number: 3924929
zbMATH Keywords: cyclotomic function field
zbMATH Keywords: divisibility of class numbers
zbMATH Keywords: irregular primes
author: Keqin Feng
MaRDI profile type: Publication
cites work: The class number of cyclotomic function fields
cites work: Kummer and Herbrand criterion in the theory of function fields
cites work: On a new type of \(L\)-function for algebraic curves over finite fields
cites work: Explicit Class Field Theory for Rational Function Fields
cites work: On the zeta function for function fields over \(F_ p\).
full work available at URL: 0022-314x(86)90071-5
OpenAlex ID: W2006352939
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description: scientific article; zbMATH DE number 3924929