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 . 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
European Journal of Mathematics – Springer Journals
Published: Jun 5, 2018
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