Let $K={𝔽}_q\left(T\right)$ be the rational function field over a finite field of $q$ elements. For any polynomial $f\left(T\right)\in {𝔽}_q\left[T\right]$ with positive degree, denote by ${\Lambda }_f$ the torsion points of the Carlitz module for the polynomial ring ${𝔽}_q\left[T\right]$. In this short paper, we will determine an explicit formula for the analytic class number for the unique subfield $M$ of the cyclotomic function field $K\left({\Lambda }_P\right)$ of degree $k$ over ${𝔽}_q\left(T\right)$, where $P\in {𝔽}_q\left[T\right]$ is an irreducible polynomial of positive degree and $k>1$ is a positive divisor of $q-1$. A formula for the analytic class...

Let $L/K$ be an extension of algebraic number fields, where $L$ is abelian over $\mathbb{Q}$. In this paper we give an explicit description of the associated order ${𝒜}_{L/K}$ of this extension when $K$ is a cyclotomic field, and prove that $o_L$, the ring of integers of $L$, is then isomorphic to ${𝒜}_{L/K}$. This generalizes previous results of Leopoldt, Chan Lim and Bley. Furthermore we show that ${𝒜}_{L/K}$ is the maximal order if $L/K$ is a cyclic and totally wildly ramified extension which is linearly disjoint to ${\mathbb{Q}}^{\left(m^{\text{'}}\right)}/K$, where $m^{\text{'}}$ is the conductor...

Let $S$ be a subset of ${\mathbf{F}}_q$, the field of $q$ elements and $h\in {\mathbf{F}}_q\left[x\right]$ a polynomial of degree $d\>1$ with no roots in $S$. Consider the group generated by the image of $\{x-s\mid s\in S\}$ in the group of units of the ring ${\mathbf{F}}_q\left[x\right]/\left(h\right)$. In this paper we present a number of lower bounds for the size of this group. Our main motivation is an application to the recent polynomial time primality testing algorithm [AKS]. The bounds have also applications to graph theory and to the bounding of the number of rational points on abelian covers of...

Let $p$ be a prime number. A finite Galois extension $N/F$ of a number field $F$ with group $G$ has a normal $p$-integral basis ($p$-NIB for short) when ${𝒪}_N^{\prime }$ is free of rank one over the group ring ${𝒪}_F^{\prime }\left[G\right]$. Here, ${𝒪}_F^{\prime }={𝒪}_F[1/p]$ is the ring of $p$-integers of $F$. Let $m=p^e$ be a power of $p$ and $N/F$ a cyclic extension of degree $m$. When ${\zeta }_m\in F^{\times }$, we give a necessary and sufficient condition for $N/F$ to have a $p$-NIB (Theorem 3). When ${\zeta }_m\notin F^{\times }$ and $p\nmid [F\left({\zeta }_m\right):F]$, we show that $N/F$ has a $p$-NIB if and only if $N\left({\zeta }_m\right)/F\left({\zeta }_m\right)$ has a $p$-NIB (Theorem 1). When $p$ divides $[F\left({\zeta }_m\right):F]$, we show that this...

We show explicit forms of the Bertini-Noether reduction theorem and of the Hilbert irreducibility theorem. Our approach recasts in a polynomial context the geometric Grothendieck good reduction criterion and the congruence approach to HIT for covers of the line. A notion of “bad primes” of a polynomial P ∈ ℚ[T,Y] irreducible over ℚ̅ is introduced, which plays a central and unifying role. For such a polynomial P, we deduce a new bound for the least integer t₀ ≥ 0 such that P(t₀,Y) is...

A (monic) polynomial $f\left(x\right)\in \mathbb{Z}\left[x\right]$ is called if the congruence $f\left(x\right)\equiv 0$ mod $m$ has a solution for all positive integers $m$. Call $f\left(x\right)$ if it is intersective and has no rational root. It was proved by the author that every finite noncyclic solvable group $G$ can be realized as the Galois group over $\mathbb{Q}$ of a nontrivially intersective polynomial (noncyclic is a necessary condition). Our first remark is the observation that the corresponding result for nonsolvable $G$ reduces to the ordinary inverse Galois...

Let K be a field and let L = K[ξ] be a finite field extension of K of degree m > 1. If f ∈ L[Z] is a polynomial, then there exist unique polynomials $u₀,...,u_{m-1}\in K[X₀,...,X_{m-1}]$ such that $f\left({\sum }_{j=0}^{m-1}{\xi }^j{X}_j\right)={\sum }_{j=0}^{m-1}{\xi }^j{u}_j$. A. Nowicki and S. Spodzieja proved that, if K is a field of characteristic zero and f ≠ 0, then $u₀,...,u_{m-1}$ have no common divisor in $K[X₀,...,X_{m-1}]$ of positive degree. We extend this result to the case when L is a separable extension of a field K of arbitrary characteristic. We also show that the same is true for a formal power series in several...