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Who is responsible for the generalization of Hamilton's quaternions to other types of quaternion algebras, and when did this occur? In particular, Hamilton's quaternions are the 4-dimensional algebra generated over $\mathbb{R}$ by the elements $1, i, j, ij$, where $i^2=j^2=-1$, and $ij=-ji$, invented in the 1840s. In a generalized quaternion algebra, we allow $\mathbb{R}$ to be replaced by any field $K$ not having characteristic 2, then we choose $a,b\in K-\{0\}$, and let $i^2=a$, $j^2=b$. The primary reference folks give for the latter version is Marie-France Vigneras' book Arithmetique Des Algebres De Quaternions, published in 1980. I would love to see the references in that book, but unfortunately it's out of print, and the (choppy) English translation I find online leaves them out.

If one follows the evolution of Hamilton's idea, one finds that Graves' invention of the octaves (later octonians) is the most directly inspired successor (but not a generalization). There is also a lesser known construction by Macfarlane where he investigates what happens when $a=b=1$ (using $a,b$ as above), dating back to 1891, but it seems that did not catch on.

Perhaps such structures were studied before they were called "quaternion algebras," since this definition also characterizes all possible central-simple 4-dimensional algebras over a field (char$\neq2$). If this is the case, I wonder who was studying them before they were given that name, and I wonder who was responsible for connecting it back to Hamilton's terminology.

I'm asking not only out of a curiosity about the history, but also because I want to see what were the motivations and intentions behind the generalization when it was first introduced, as compared to current applications.

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Strictly speaking, your concrete description of quaternion algebras over "any" field $K$ requires $K$ not to have characteristic 2. If you want to describe quaternion algebras in characteristic 2 by means of formulas, then you could use $i^2 + i = a$, $j^2 = b$, and $ji = (i+1)j$ where $a \in K$ and $b \in K - \{0\}$. –  KConrad Feb 1 at 21:54
You're quite right, I'm correcting the question to reflect this (I'm okay with ignoring the characteristic 2 case). –  j0equ1nn Feb 1 at 23:40
If this is inappropriate to post here, someone please let me know, but for anyone in the New York City area interested in quaternion algebras, consider checking out my seminar at the CUNY Graduate Center: quaternion.ws.gc.cuny.edu –  j0equ1nn Feb 25 at 0:13
If that post leads someone to come to the seminar, I'd be interested to know that. :) –  KConrad Feb 25 at 0:23
Haha yeah me too. It's been pretty hard finding quaternion enthusiasts locally, even in NYC, so I guess I'm advertising wherever I think of it. –  j0equ1nn Mar 3 at 19:02
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1 Answer

up vote 24 down vote accepted

In the early 1900s, Dickson introduced what he called generalized quaternion algebras over any field $K$ of characteristic not 2. These are exactly what we'd call quaternion algebras over $K$. His definition was in terms of a basis with rules for their products, and he gave a criterion for these to be division rings. In particular, these were the first noncommutative division rings besides the quaternions of Hamilton, aside from subrings of Hamilton's quaternions.

Three of Dickson's works where he introduces these algebras are

(1) Linear Algebras, Trans. AMS ${\bf 13}$ (1912) 59-73.

(2) Linear Associative Algebras and Abelian Equations, Trans. AMS ${\bf 15}$ (1914), 31-46.

(3) Algebras and Their Arithmetics, Univ. of Chicago Press, 1923.

In (1) he gives the defining equations for a generalized quaternion algebra and the norm criterion for it to be a division algebra (pp. 65-66), though without using the label "generalized quarernion algebra". He writes in a footnote that this work goes back to 1906.

In (2) he constructs cyclic algebras, without using that name, and calls the special case of dimension 4 a generalized quaternion algebra. (Archaic terminology alert: Dickson refers to equations defining cyclic Galois extensions as uniserial abelian equations.)

In (3) he defines cyclic algebras again without using that name(p. 65), writes them as $D$, refers to them as algebras of type $D$, and remarks in a footnote on p. 66 that Wedderburn calls them Dickson algebras. Near the end of the book he looks at generalized quaternion algebras over the rationals with $\mathbf Q(i)$ as a maximal subfield, and as a particular example he uses the Hamilton quaternions over $\mathbf Q$ to describe all rational and integral solutions of certain quadratic Diophantine equations in several variables. Think about how a sum of four squares factors over the quaternions to imagine how arithmetic properties of quaternions could be useful to analyze a Diophantine equation involving a sum of four squares; this is similar in spirit to the way the Gaussian integers are useful in studying a Diophantine equation involving a sum of two squares. The term "arithmetics" in the title of the book is, as far as I can tell, Dickson's label for what we'd call maximal orders, so the book would be called today "Algebras and Their Maximal Orders".

The motivation for Dickson's interest in quaternion algebras was the earlier development of integral Hamilton quaternions, due first to Lipschitz (all integral coefficients, which is clunky in the same way that $\mathbf Z[\sqrt{-3}]$ is compared to $\mathbf Z[(1+\sqrt{-3})/2]$) and then to Hurwitz (all integral or all half-integral coefficients). To study quadratic Diophantine equations in several variables going beyond a sum of four squares, Dickson was led to extend the original definition of quaternions. His main interest was developing the right theory of generalized integral quaternions, rather than just a theory over a field. A further reference in this direction is Dickson's paper

(4) On the theory of numbers and generalized quaternions, Amer. J. Math ${\bf 46}$ (1924), 1-16.

Concerning the connection to central simple algebras, Dickson was the first to show any division algebra that is 4-dimensional over its center is cyclic, at least outside characteristic 2 since he didn't have a good definition in characteristic 2. Taking into account Wedderburn's theorem that every finite-dimensional central simple algebra over a field is a matrix algebra over a division algebra, and that matrix algebras are cyclic, Dickson had shown that every 4-dimensional CSA over a field not having characteristic 2 is an algebra of "his" type. (Wedderburn proved the analogous theorem for dimension 9.)

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Thanks! You've helped me in the past to understand mathematical aspects of quaternion algebras, now you've helped me with their history as well -- much appreciated! –  j0equ1nn Feb 1 at 23:48
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