Archive for the ‘Algebra’ Category
We show how Partial Fraction Decomposition is a consequence of the Chinese Remainder Theorem. Our results hold for arbitrary principal ideal domains, with stronger results for certain Euclidean rings. Therefore, we do not only have Partial Fraction Decomposition for rational function fields, but also for the rational numbers.
This short post discusses a cute statement which shows the power of a certain subset of axioms of a ring, including prominently the distributive law.
We show how to prove a number theoretic inequality, originating from geometry, using an elementary approach.
This post shows a diagram, listing a lot of inequalities and showing implications between them.
In this post, we consider the quest of computing the 5-adic expansion of 1/2. We begin with introducing p-adic integers and numbers, and discussing when certain polynomials with coefficients in the integers have zeroes in the p-adic integers. This question is closely related to Hensel’s lemma, which can be proven using an algebraic version of Newton’s iteration. We use this to compute approximations of rational numbers in the p-adics, and consider which p-adic numers have an eventually periodic expansion.
We compare the tasks of finding points of a lattice, computing the structure of finite abelian groups and explaining algorithms. We show up relations between these three topics and, as an example, depict the baby-step giant-step algorithm for order computation, as well as Terr’s modification of this algorithm.
A standard topic in linear algebra is the dual space of a vector space, as well as the canonical embedding of a vector space in its double dual. Moreover, transposition of homomorphisms in terms of dual spaces is rather well known. Something less known is that one has a canonical map from the dual of V tensored with W to the space of homomorphisms from V to W. In this abstract nonsense post, we describe these canonical maps, their interplay, and try to determine their images.
We consider the property of an n times n matrix of being diagonalizable. Is this property open in the standard topology, or the Zariski topology? The emphasis lies on the real and complex numbers, as well as on arbitrary algebraically closed fields.
We will discuss Euclidean domains together with a constructive proof of the fact that every two elements have a greatest common divisor, which is essentially the Euclidean algorithm.
We will state several (more or less) useful properties of the Extended Euclidean Algorithm, in particular for the case of integers and univariate polynomials over a field.
Following a suggestion by A. Maevskiy, we show how the Hasse derivative can be extended to partial Hasse derivative in arbitrary multivariate polynomial rings. We show multivariate versions of Taylor’s Formula, of the Identity Theorem, and of the Generalized Leibnitz Rule.