Cubic fields: a primer

Cubic fields: a primer European Journal of Mathematics https://doi.org/10.1007/s40879-018-0258-5 RESEARCH ARTICLE Cubic fields: a primer 1 2 Sophie Marques · Kenneth Ward Received: 19 April 2017 / Revised: 31 October 2017 / Accepted: 12 May 2018 © Springer International Publishing AG, part of Springer Nature 2018 Abstract We classify all cubic extensions of any field of arbitrary characteristic, up to isomorphism, via an explicit construction involving three fundamental types of cubic forms. This classification is simple, in the sense that it gives a short and explicit algorithm for passing from an arbitrary cubic into one of our three classes of cubics. We also deduce a classification of any Galois cubic extension of a field. Keywords Cyclotomy · Cubic · Function field · Finite field · Galois Mathematics Subject Classification 11T22 · 11R32 · 11R16 · 11T55 · 11R58 1 Introduction In this paper, we give a complete classification of cubic field extensions up to isomor- phism over an arbitrary field of any characteristic, which we had begun in [4]. This classification enjoys special benefits, including that it has not been done before. For cubic extensions of global fields, this classification allows one to read off essential Funding was provided by American University CAS http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png European Journal of Mathematics Springer Journals

Cubic fields: a primer

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Publisher
Springer International Publishing
Copyright
Copyright © 2018 by Springer International Publishing AG, part of Springer Nature
Subject
Mathematics; Algebraic Geometry
ISSN
2199-675X
eISSN
2199-6768
D.O.I.
10.1007/s40879-018-0258-5
Publisher site
See Article on Publisher Site

Abstract

European Journal of Mathematics https://doi.org/10.1007/s40879-018-0258-5 RESEARCH ARTICLE Cubic fields: a primer 1 2 Sophie Marques · Kenneth Ward Received: 19 April 2017 / Revised: 31 October 2017 / Accepted: 12 May 2018 © Springer International Publishing AG, part of Springer Nature 2018 Abstract We classify all cubic extensions of any field of arbitrary characteristic, up to isomorphism, via an explicit construction involving three fundamental types of cubic forms. This classification is simple, in the sense that it gives a short and explicit algorithm for passing from an arbitrary cubic into one of our three classes of cubics. We also deduce a classification of any Galois cubic extension of a field. Keywords Cyclotomy · Cubic · Function field · Finite field · Galois Mathematics Subject Classification 11T22 · 11R32 · 11R16 · 11T55 · 11R58 1 Introduction In this paper, we give a complete classification of cubic field extensions up to isomor- phism over an arbitrary field of any characteristic, which we had begun in [4]. This classification enjoys special benefits, including that it has not been done before. For cubic extensions of global fields, this classification allows one to read off essential Funding was provided by American University CAS

Journal

European Journal of MathematicsSpringer Journals

Published: Jun 5, 2018

References

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