Let (K, v) be a Henselian valued field. In this article, we use Okutsu sequences for monic, irreducible polynomials in K[x], and their relationship with MacLane chains of inductive valuations on to obtain some results the computation invariants algebraic elements over K.

Our main result is that any topological algebra based on a Boolean space is the extended Stone dual space of a certain associated Boolean algebra with additional operations. A particular case of this result is that the profinite completion of any abstract algebra is the extended Stone dual space of the Boolean algebra of recognisable subsets of the abstract algebra endowed with certain residuat...

Abstract For a subring $R$ of the rational numbers, we study -localization functors in local homotopy theory simplicial presheaves on small site and then ${\mathbb {A}}^1$ -homotopy theory. To this end, introduce analyze two notions nilpotence for spaces theory, paying attention to future applications vector bundles. We show that behaves controlled fashion nilpotent consider. classifying space ...

Given a knot K in the 3-sphere, let QK be its fundamental quandle as introduced by D. Joyce. Its first homology group is easily seen to be H1(QK) ∼= Z. We prove that H2(QK) = 0 if and only if K is trivial, and H2(QK) ∼= Z whenever K is non-trivial. An analogous result holds for links, thus characterizing trivial components. More detailed information can be derived from the conjugation quandle: ...

Let p:E B be a fibration over a connected space B with fiber F. The Eilenberg-Moore spectral sequence of p is a second quadrant spectral sequence which tries and sometimes fails to converge strongly to the homology of F (see 1-53). The purpose of this paper is to determine what the spectral sequence does converge to. An abstract answer (Theorem 1.1) is that the spectral sequence almost always c...