Field (mathematics)

Summary

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and p-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements. The relation of two fields is expressed by the notion of a field extension. Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions. Among other results, this theory shows that

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Teaching Unit in Geotechnologies for Mapping Changes of the Built and Natural Environment

Olivier Feihl, Pierre-Yves Gilliéron, Geoffrey Vincent, Jérôme Zufferey

Teaching geomatics is a continuous challenge with the evolution of technologies which enables more and more accurate measurement and modelling of the environment. This paper will present an original pedagogical approach based on a teaching unit gathering students in architecture, civil an environmental engineering. Working together in collecting and visualizing data from the built and natural environment with advanced tools like laser scanners is a very stimulating approach that requires multiple competences in mapping and data analysis. A number of practical examples (bridge maintenance, archaeological site, city management) are presented from the field operations to the 3D visualisation and mapping. Each of these topics is illustrated with comparison maps.

2015

Bounds for the Euclidean minima of function fields

Piotr Aleksander Maciak

In this paper, we define Euclidean minima for function fields and give some bound for this invariant. We furthermore show that the results are analogous to those obtained in the number field case. (C) 2013 The Authors. Published by Elsevier Inc. All rights reserved.

Academic Press Inc Elsevier Science2014

Direct Observation of Magnon Fractionalization in the Quantum Spin Ladder

Daniel Biner, Jean-Sebastien Caux, Joël Mesot, Bruce Normand, Henrik Moodysson Rønnow, Christian Rüegg

We measure by inelastic neutron scattering the spin excitation spectra as a function of applied magnetic field in the quantum spin-ladder material (C5H12N)(2)CuBr4. Discrete magnon modes at low fields in the quantum disordered phase and at high fields in the saturated phase contrast sharply with a spinon continuum at intermediate fields characteristic of the Luttinger-liquid phase. By tuning the magnetic field, we drive the fractionalization of magnons into spinons and, in this deconfined regime, observe both commensurate and incommensurate continua.

2009