We present a fast parallel deterministic algorithm for testing multivariate integral polynomials for absolute irreducibility, that is irreducibility over the complex numbers. More precisely, we establish that the set of absolutely irreducible integral polynomials belongs to the complexity class NC of Boolean circuits of polynomial size and logarithmic depth. Therefore it also belongs to the cla...

In constructive algebra one cannot in general decide the irreducibility of a polynomial over a field K. This poses some problems to showing the existence of the algebraic closure of K. We give a possible constructive interpretation of the existence of the algebraic closure of a field in characteristic 0 by building, in a constructive metatheory, a suitable site model where there is such an alge...

The notion of subdirect irreducibility in the context of languages without equality, as presented by Elgueta, is extended in order to obtain subdirect representation theorems for abstract and reduced classes of structure systems. Structure systems serve as models of firstorder theories but, rather than having universal algebras as their algebraic reducts, they have algebraic systems in the sens...

This paper presents a criterion for testing the irreducibility of a polynomial over an algebraic extension field. Using this criterion and the characteristic set method, the authors give a criterion for testing whether certain difference ascending chains are strong irreducible, and as a consequence, whether the saturation ideals of these ascending chains are prime ideals.

We define the decomposable extensions of difference fields and study the irreducibility of q-Painlevé equation of type A 7 ′ . Every strongly normal extension or Liouville-Franke extension, the latter of which is a difference analogue of the Liouvillian extension, satisfies that its appropriate algebraic closure is a decomposable extension.

We find geometric and arithmetic conditions in order to characterize the irreducibility of the determinant of the generic Vandermonde matrix over the algebraic closure of any field k. We also characterize those determinants whose tropicalization with respect to the variables of a row is irreducible.

We find geometric and arithmetic conditions on a finite set of integer exponents and the characteristic of a field k in order to characterize the irreducibility of the determinant of the generic Vandermonde matrix determined by these exponents in the algebraic closure of k. Tools from Tropical Geometry are then applied in order to characterize those determinants that are irreducible over any fi...

Let L/Qp be a finite extension, and let G be a locally L-analytic group such as the L-points of an algebraic group over L. The categories of smooth and of finite-dimensional algebraic representations of G may be viewed as subcategories of the category of continuous representations of G in locally convex L-vector spaces. This larger category contains many interesting new objects, such as the act...

3 Topics in Commutative Algebra 2 3.1 Rings and Fields . . . . . . . . . . . . . . . . . . . . . . . . . 2 3.2 Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.3 Quotient Rings and Homomorphisms . . . . . . . . . . . . . . 5 3.4 The Characteristic of a Ring . . . . . . . . . . . . . . . . . . . 7 3.5 Polynomial Rings in Several Variables . . . . . . . . . . . . . . 7 3.6...