European Journal of Mathematics
Cubic ﬁelds: a primer
· 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
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 Mellon Fund.
Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch
7701, South Africa
Department of Mathematics and Statistics, American University, 4400 Massachusetts Avenue NW,
Washington, DC 20016, USA