Related papers: An explicit triangular integral basis for any sepa…

We construct explicit exponential bases on triangles in R^2 and on infinite unions of segments on the real line.

We define a variant of normal basis, called a {\em Galois scaffolding}, that allows for an easy determination of valuation, and has implications for Galois module structure. We identify fully ramified, elementary abelian extensions of local…

We prove that function fields of varieties of dimension at least two over an algebraic closure of a finite field are determined, modulo purely inseparable extensions, by the quotient by the second term in the lower central series of their…

We provide a characterization of infinite algebraic Galois extensions of the rationals with uniformly bounded local degrees, giving a detailed proof of all the results announced in a paper by Checcoli and Zannier and obtaining relevant…

We give an elementary construction of an arbitrary differentially closed field and of a universal differential extension of a differential field in terms of Nash function fields. We also give a characterization of any Archimedean ordered…

This paper justifies an assertion in (Elder, Proc AMS 137 (2009), no 4, 1193--1203) that Galois scaffolds make the questions of Galois module structure tractable. Let $k$ be a perfect field of characteristic $p$ and let $K=k((T))$. For the…

We prove a Galois-type correspondence between compositions of purely inseparable field extensions (including infinite ones) and subalgebras of differential operators. This correspondence can be utilized to establish a connection between…

A method to construct trihamiltonian extensions of a separable system is presented. The procedure is tested for systems, with a natural Hamiltonian, separable in classical sense in one of the four orthogonal separable coordinate systems of…

We extend the theorem of Liouville on integration in finite terms to include dilogarithmic integrals. The results provide a necessary and sufficient condition for an element of the base field to have an antiderivative in a field extension…

We construct a constrained trivariate extension of the univariate normalized B-basis of the vector space of trigonometric polynomials of arbitrary (finite) order n defined on any compact interval [0,\alpha], where \alpha is a fixed (shape)…

Transcendental Liouvillian extensions are differential fields, in which one can model poly-logarithmic, hyperexponential, and trigonometric functions, logarithmic integrals, and their (nested) rational expressions. For such an extension…

Using the action of the Galois group of a normal extension of number fields, we generalize and symmetrize various fundamental statements in algebra and algebraic number theory concerning splitting types of prime ideals, factorization types…

We determine the absolute differential Galois group of the field $\mathbb{C}(x)$ of rational functions: It is the free proalgebraic group on a set of cardinality $|\mathbb{C}|$. This solves a longstanding open problem posed by B.H. Matzat.…

We obtain new complexity bounds for computing a triangular integral basis of a number field or a function field. We reach for function fields a softly linear cost with respect to the size of the output when the residual characteristic is…

We construct explicit examples of cubic surfaces over $\bbQ$ such that the 27 lines are acted upon by the index two subgroup of the maximal possible Galois group. This is the simple group of order $25 920$. Our examples are given in…

To generalize the notion of Galois closure for separable field extensions, we devise a notion of $G$-closure for algebras of commutative rings $R\to A$, where $A$ is locally free of rank $n$ as an $R$-module and $G$ is a subgroup of…

A Galois scaffold, in a Galois extension of local fields with perfect residue fields, is an adaptation of the normal basis to the valuation of the extension field, and thus can be applied to answer questions of Galois module structure. Here…

Let $K$ be an imaginary quadratic field with discriminant $d_K\leq-7$. We deal with problems of constructing normal bases between abelian extensions of $K$ by making use of singular values of Siegel functions. First, we show that a…

We introduce the universal unitarily graded A-algebra for a commutative ring A and an arbitrary abelian extension U of the group of units of A, and use this concept to give simplified proofs of the main theorems of co-Galois theory in the…

Given an arbitrary field $F$, we describe all Galois extensions $L/F$ whose Galois groups are isomorphic to the group of upper triangular unipotent 4-by-4 matrices with entries in the field of two elements.