# Why did Jordan introduce his canonical form?

Camille Jordan's famous canonical form for matrices over algebraically closed fields, is considered an important result nowadays, commonly taught to all students of mathematics in undergraduate linear algebra courses. The vast majority of them only get to see it applied in one case: in the study of linear systems of differential equations with constant coefficients, where it is used very effectively to solve all such systems. Other applications that are sometimes seen are its use in systems of linear recurrence relations, and in Markov chains, but that's about it.

However, originally, Jordan did not discover his form to solve any of these problems - it was, in fact, a problem in Galois theory that motivated his research. Given that I cannot read french, and Jordan's Traité was never translated to english, I give the only english reference to it that I could find, from Hawkins' "The Mathematics of Frobenius in Context" (2013), pages 137-138:

Jordan... preferred to make the consideration of homogeneous linear substitutions fundamental, and in a paper of 1867, he indicated their important role in the problem of determining all the irreducible equations of a given degree that are solvable by radicals. In connection with this problem he sought, in a paper of 1868, to determine the solvable subgroups of the group of linear substitutions in two variables... To do it he used the fact that by a linear change of variables, a linear substitution S could be put in one of a limited number of “canonical forms” depending on the nature of the roots of det(S−kI) ≡ 0 (mod p). His method of constructing solvable subgroups was to build them up from their composition series, and this involved determining all linear substitutions that commute with a given substitution S. To this end, he introduced the possible canonical forms for S.

Hawkins then mentions that Jordan further generalized his theorem to linear substitutions in n variables, and that a few years later he realized the relevance of his theorem to systems of differential equations.

Two points are still unclear to me. First, from this summary it seems that Jordan was interested in matrices (here termed linear substitutions) over finite fields - given that all the work is done mod p. Finite fields, however, are never algebraically complete - so which theorem did Jordan even prove? The usual theorem taught is not applicable here. Second, this summary explains what is the problem that Jordan was interested in, but only sketches very roughly how he solved it. How exactly is this classification of matrices (up to similarity) relevant for determining which irredicuble polynomials are solvable?

• Jordan was not the first to "discover" it, for detailed history see Weierstrass and the theory of matrices by Hawkins. Jordan's reasoning in the Traite is described on pp. 144-46. – Conifold Mar 14 '20 at 5:35
• Section 14.3 of Cox’s “Galois Theory”, such as Corollary 14.3.18 and Prop. 14.3.20, give context for linking solvability of polynomials and matrix groups mod $p$. – KCd Jan 25 at 5:41