In function fields in positive characteristic, we provide a concrete example of completely normal elements for a finite Galois extension. More precisely, for a nonabelian extension, we construct completely normal elements for Drinfeld modular function fields using Siegel functions in function fields. For an abelian extension, we construct completely normal elements for cyclotomic function fields.

We describe an algorithm for finding the coefficients of $F(X)$ modulo powers of $p$ , where $p\neq 2$ is a prime number and $F(X)$ is the power series associated to the zeta function of Kubota and Leopoldt. We next calculate the 5-adic and 7-adic ${\it\lambda}$ -invariants attached to those cubic extensions $K/\mathbb{Q}$ with cyclic Galois group ${\mathcal{A}}_{3}$ (up to field discriminant ${<}10^{7}$ ), and also tabulate the class number of $K(e^{2{\it\pi}i/p})$ for $p=5$ and $p=7$ . If the ${\it\lambda}$ -invariant is greater than zero, we then determine all the zeros for the corresponding branches of the $p$ -adic $L$ -function and deduce ${\rm\Lambda}$ -monogeneity for the class group tower over the cyclotomic $\mathbb{Z}_{p}$ -extension of $K$ .

Supplementary materials are available with this article.

We define a topological space over the p-adic numbers, in which Euler products and Dirichlet series converge. We then show how the classical Riemann zeta function has a (p-adic) Euler product structure at the negative integers. Finally, as a corollary of these results, we derive a new formula for the non-Archimedean Euler–Mascheroni constant.

We prove that the p-adic zeta-function constructed by Kubota and Leopoldt has the Dirichlet series expansion Where the convergence of the first summation is for the p-adic topology. The proof of this formula relates the values of p(–s, ω1+σ) for s ∈ Zp, with a branch of the ‘sth-fractional derivative’, of a suitable generating function.

In this paper, we generalize the Kučera's group-determinant formulae to obtain the real and relative class number formulae of any subfield of cyclotomic function fields with arbitrary conductor in the form of a product of determinants. From these formulae, we generalize the relative class number formula of Rosen and Bae-Kang and obtain analogous results of Tsumura and Hirabayashi for an intermediate field in the tower of cyclotomic function fields with prime power conductor.

We find a basis for the universal punctured even distribution and then a basis for the cyclotomic units over function fields.