The characteristic polynomial and determinant are not ad hoc constructions

by Skip Garibaldi
Published in American Mathematical Monthly 111 #9 (2004), 761--778
math.RA/0203276

There are good definitions of the determinant of an n-by-n matrix: as the factor by which it distorts volumes, or by the the way it acts on the n-th exterior power of the vector space. The definitions one finds in a an undergraduate linear algebra book are fine for that purpose, but seem totally ad hoc to me.

But even the two "good" definitions share a flaw. There are analogues of the determinant and characteristic polynomial for quaternions, octonions, Jordan algebras, finite-dimensional field extensions, etc., and the two "good" definitions do not cover those cases.

This paper gives a definition that handles all the cases simultaneously, including deriving the usual formulas for the determinant and characteristic polynomials of matrices from the general definition. This note is intended for a broad audience; the only background required is one year of graduate algebra.

Version of 30 April 04. It is very close to the published version.
(dvi), (dvi.gz), (ps), (ps.gz), (pdf), (pdf.gz)

The general notion of "characteristic polynomial" given in my paper above has been known about for a long time, sometimes under the name "generic minimum polynomial". (The interesting part of my paper is that I derive the traditional---and seemingly ad hoc---linear algebra definitions of determinant and characteristic polynomial from the general notion.) In addition to the references in my paper, the following also discuss the characteristic polynomial:

O. Loos, Generically algebraic Jordan algebras over commutative rings, preprint, February 2005.
R.H. Oehmke, On the generic polynomial of an algebra, Scripta Mathematica 29 (1973), #3-4, 331-336.
J. Tits, A theorem on generic norms of strictly power associative algebras, Proc. Amer. Math. Soc. 15 (1964), 35-36. (article on JSTOR)
J. Dieudonné, Sur le polyôme principal d'une algèbre, Arch. Math. 8 (1957), 81-84.

The coefficient c₂ (in the notation of my paper) of the characteristic polynomial defines a quadratic form on the algebra. In characteristic 2, it seems to be a substitute for the quadratic form derived from the trace (c₁ in my paper). Here are some papers concerning that form:

A.-M. Bergé and J. Martinet, Formes quadratiques et extension en caractéristique 2, Annales de l'institut Fourier 35 (1985), 57-77.
T. Unger, A note on surrogate forms of central simple algebras, Mathematical Proceedings of the Royal Irish Academy 101A (2001), 125-135.
G. Berhuy and C. Frings, On the second trace form of central simple algebras in characteristic two, manuscripta math. 106 (2001), 1-12.
H.P. Petersson, Structure theorems for Jordan algebras of degree three over fields of arbitrary characteristic, Comm. Algebra 32 (2004), 1019-1049.

One can find a similar notion of characteristic polynomial in the literature (e.g., in Bourbaki and in Section 10 of Hilbert's Zahlbericht), where instead of specializing the minimal polynomial of the generic element, one specializes the characteristic polynomial of left multiplication by the generic element. This definition gives different answers than the one in my paper, see for example my Remark 5.4.