In this paper we give a procedure to search for prime divisors of class numbers of real abelian fields and present a table of odd primes < 10000 not dividing the degree that divide the class numbers of fields of conductor ≤ 2000. Cohen–Lenstra heuristics allow us to conjecture that no larger prime divisors should exist. Previous computations have been largely limited to prime power conductors.

We study a family of quintic polynomials discoverd by Emma Lehmer. We show that the roots are fundamental units for the corresponding quintic fields. These fields have large class numbers and several examples are calculated. As a consequence, we show that for the prime p = 641491 the class number of the maximal real subfield of the pth cyclotomic field is divisible by the prime 1566401. In an a...

We determine information sets for the generalized Reed-Muller codes and use these to apply partial permutation decoding to codes from finite geometries over prime fields. We also obtain new bases of minimum-weight vectors for the codes of the designs of points and hyperplanes over prime fields.

Many observations about coalgebras were inspired by comparable situations for algebras. Despite the prominent role of prime algebras, the theory of a corresponding notion for coalgebras was not well understood so far. Coalgebras C over fields may be called coprime provided the dual algebra C∗ is prime. This definition, however, is not intrinsic it strongly depends on the base ring being a field...

In this brief note we connect the discrete logarithm problem over prime fields in the safe prime case to the logarithmic derivative.

In IEEE P1363, two kinds of finite fields, “Prime Finite Fields” and “Characteristic Two Finite Fields” have been standardized. We propose “Optimal Extension Fields (OEF)” in addition to the two fields. OEF is efficient to compute [1–3].

S. Chowla conjectured that every prime p has the property that there are infinitely many imaginary quadratic fields whose class number is not a multiple of p. Gauss’ genus theory guarantees the existence of infinitely many such fields when p = 2, and the work of Davenport and Heilbronn [D-H] suffices for the prime p = 3. In addition, the DavenportHeilbronn result demonstrates that a positive pr...

We present an implementation of elliptic curves and of hyperelliptic curves of genus 2 and 3 over prime fields. To achieve a fair comparison between the different types of groups, we developed an ad-hoc arithmetic library, designed to remove most of the overheads that penalize implementations of curve-based cryptography over prime fields. These overheads get worse for smaller fields, and thus f...

In this paper, we employ the concept of the Generalized Discrete Fourier Transform, which in turn relies on the Hasse derivative of polynomials, to give a general construction of Reed-Solomon codes over Galois fields of characteristic not necessarily co-prime with the length of the code. The constructed linear codes  enjoy nice algebraic properties just as the classic one.