Related papers: Explicit genus formula for any separable cubic glo…
We classify all cubic extensions of any field of arbitrary characteristic, up to isomorphism, via an explicit construction involving three fundamental types of cubic forms. We deduce a classification of any Galois cubic extension of a…
Let $k$ be a rational congruence function field and consider an arbitrary finite separable extension $K/k$. If for each prime in $k$ ramified in $K$ we have that at least one ramification index is not divided by the characteristic of $K$,…
We determine an explicit triangular integral basis for any separable cubic extension of a rational function field over a finite field in any characteristic. We obtain a formula for the discriminant of every such extension in terms of a…
In this paper we obtain the extended genus field of a global field. First we define the extended genus field of a global function field and we obtain, via class field theory, the description of the extended genus field of an arbitrary…
In this paper we obtain the extended genus field of a finite abelian extension of a global rational function field. We first study the case of a cyclic extension of prime power degree. Next, we use that the extended genus fields of a…
In this paper we first obtain the genus field of a finite abelian non-Kummer $l$--extension of a global rational function field. Then, using that the genus field of a composite of two abelian extensions of a global rational function field…
In this paper we obtain the genus field of a general Kummer extension of a global rational function field. We study first the case of a general Kummer extension of degree a power of a prime. Then we prove that the genus field of a composite…
In this paper we present an algorithm that computes the genus of a global function field. Let F/k be function field over a field k, and let k0 be the full constant field of F/k. By using lattices over subrings of F, we can express the genus…
We classify all cubic function fields over any finite field, particularly developing a complete Galois theory which includes those cases when the constant field is missing certain roots of unity. In doing so, we find criteria which allow…
In this paper we find the genus field of finite abelian extensions of the global rational function field. We introduce the term conductor of constants for these extensions and determine it in terms of other invariants. We study the…
In this work we present some arithmetic properties of families of abelian $p$--extensions of global function fields, among which are their generators and their type of ramification and decomposition.
A formula for calculating Extensions of (mainly integral) Polynomial Functors is established, based upon projective resolutions. Sample computations are performed, which, in particular, exhibit a surprising non-trivial extension of Divided…
This paper contains an account of arbitrary cubic function fields of characteristic three. We define a standard form for an arbitrary cubic curve and consider its function field. By considering an integral basis for the maximal order of…
We compute the Galois group of the splitting field $F$ of any irreducible and separable polynomial $f(x)=x^6+ax^3+b$ with $a,b\in K$, a field with characteristic different from two. The proofs require to distinguish between two cases:…
Our main result is that any real cubic algebraic number has a continued fraction expansion with polynomial coefficients. Some generalizations are mentioned.
This article describes cubic function fields $L/K$ with prescribed ramification, where $K$ is a rational function field. We give general equations for such extensions, an explicit procedure to obtain a defining equation when the purely…
In the present work we give the construction of the genus field and the extended genus field of an elementary abelian $l$-extension of a field of rational functions, where $l$ is a prime number. In the Kummer case, if $K$ is contained in a…
We give an algorithm that constructs a minimal set of polynomials defining all extension of a $(\pi)$-adic field with given, inertia degree, ramification index, discriminant, ramification polygon, and residual polynomials of the segments of…
We show that, for any fixed genus $g$, the ordinary generating function for the genus $g$ partitions of an $n$-element set into $k$ blocks is algebraic. The proof involves showing that each such partition may be reduced in a unique way to a…
To any real rational function with generic ramification points we assign a combinatorial object, called a garden, which consists of a weighted labeled directed planar chord diagram and of a set of weighted rooted trees each corresponding to…