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The aim of Essential Number Theory is to publish articles of excellent utility and clarity, in all areas of number theory.

Number theory --- Number theory. --- Number study --- Numbers, Theory of --- Algebra

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This self-contained and comprehensive textbook of algebraic number theory is useful for advanced undergraduate and graduate students of mathematics. The book discusses proofs of almost all basic significant theorems of algebraic number theory including Dedekind's theorem on splitting of primes, Dirichlet's unit theorem, Minkowski's convex body theorem, Dedekind's discriminant theorem, Hermite's theorem on discriminant, Dirichlet's class number formula, and Dirichlet's theorem on primes in arithmetic progressions. A few research problems arising out of these results are mentioned together with the progress made in the direction of each problem. Following the classical approach of Dedekind's theory of ideals, the book aims at arousing the reader's interest in the current research being held in the subject area. It not only proves basic results but pairs them with recent developments, making the book relevant and thought-provoking. Historical notes are given at various places. Featured with numerous related exercises and examples, this book is of significant value to students and researchers associated with the field. The book also is suitable for independent study. The only prerequisite is basic knowledge of abstract algebra and elementary number theory. .

Number theory --- getallenleer

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Algebraic number theory. --- Number theory. --- Number study --- Numbers, Theory of --- Algebra --- Number theory --- Teoria algebraica de nombres

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This innovative undergraduate textbook approaches number theory through the lens of abstract algebra. Written in an engaging and whimsical style, this text will introduce students to rings, groups, fields, and other algebraic structures as they discover the key concepts of elementary number theory. Inquiry-based learning (IBL) appears throughout the chapters, allowing students to develop insights for upcoming sections while simultaneously strengthening their understanding of previously covered topics. The text is organized around three core themes: the notion of what a "number" is, and the premise that it takes familiarity with a large variety of number systems to fully explore number theory; the use of Diophantine equations as catalysts for introducing and developing structural ideas; and the role of abstract algebra in number theory, in particular the extent to which it provides the Fundamental Theorem of Arithmetic for various new number systems. Other aspects of modern number theory - including the study of elliptic curves, the analogs between integer and polynomial arithmetic, p-adic arithmetic, and relationships between the spectra of primes in various rings - are included in smaller but persistent threads woven through chapters and exercise sets. Each chapter concludes with exercises organized in four categories: Calculations and Informal Proofs, Formal Proofs, Computation and Experimentation, and General Number Theory Awareness. IBL "Exploration" worksheets appear in many sections, some of which involve numerical investigations. To assist students who may not have experience with programming languages, Python worksheets are available on the book's website. The final chapter provides five additional IBL explorations that reinforce and expand what students have learned, and can be used as starting points for independent projects. The topics covered in these explorations are public key cryptography, Lagrange's four-square theorem, units and Pell's Equation, various cases of the solution to Fermat's Last Theorem, and a peek into other deeper mysteries of algebraic number theory. Students should have a basic familiarity with complex numbers, matrix algebra, vector spaces, and proof techniques, as well as a spirit of adventure to explore the "numberverse.".

Number theory --- Algebra --- algebra --- getallenleer

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Previously, English used to distinguish between singular and plural second person pronouns. However, this distinction was lost during its transition into Modern English, which saw the establishment of the single form you for both singular and plural reference.Despite this, many dialects of English have always continued to explicitly mark number on second person pronouns by resorting to different linguistic strategies, both morphological and analytic. Among such morphological variants is yous, together with a host of different spelling variants such as youse, yiz, and yez, among others.This work is a synchronic, corpus-based investigation of second person plural forms in 20 varieties of English. The corpus under study here (GloWbe) contains 1.9 billion words collected on the web in 2013 and was analysed in order to uncover the usage trends of second person plural forms in present-day English. The picture that emerges displays the diatopic distribution of the forms, in addition to markers of politeness and empathy, singular-reference emphatic markers, attention-getting devices, and possessive determiner, and their frequencies of occurrence. The book pays particular attention to the fundamental function of second person plural forms in the creation and management of the speaker-hearer relationship.

English language --- Grammar --- Number --- Dialects