A note on irregular prime polynomials in cyclotomic function field theory
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.
scientific article; zbMATH DE number 3924929