# Who first noticed that the Hilbert symbol is a Steinberg symbol ?

Hilbert reformulated the quadratic reciprocity law of Gauß as a product formula $$\prod_v(a,b)_v=1$$ for the various local Hilbert symbols. For each place $v$ of $\bf Q$, the Hilbert symbol $(\ ,\ )_v$ is a bimultiplicative map $${\bf Q}_v^\times\times{\bf Q}_v^\times\to{\bf Z}^\times$$ so that, by definition, $(a,b)_v=1$ if and only if $a\in {\rm Im\;} N_b$ where $N_b$ denotes the norm map ${\bf Q}_v(\sqrt b)^\times\to{\bf Q}_v^\times$. An important property of the Hilbert symbol is that $$a+b=1\Longrightarrow (a,b)_v=1,$$ which makes it a Steinberg symbol. This property in not listed in older books such as Hasse's Number theory but it can be found in all modern treatments, such as Serre's Course in arithmetic or his Local fields, or Milnor's K-theory.

I'm curious as to who first noticed that the Hilbert symbol is a Steinberg symbol. Was it Steinberg himself ? A precise reference will be appreciated.

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Wasn't the symbol classically defined to be 1 iff the form axx+byy represents 1 in the local field? So everybody noticed it. –  paul Monsky Aug 29 '12 at 15:38
Hilbert himself defined it in terms of local norms from quadratic extensions, although he avoided the explicit mention of local fields. He derived the usual formulae for the symbol, so he (or others) could have easily verified the Steinberg property. But neither Hilbert nor Hasse mentions this property, even as a curiosity. My question is about making it explicit, which Serre does in his Corps locaux. –  Chandan Singh Dalawat Aug 29 '12 at 16:29

EDIT: After looking into the history more closely, I think it's fairly certain that the correct answer to the question is Calvin Moore. (See my added text below.)

The question is interesting and looks straightforward, but it may not have a simple answer. A number of independent lines of research, differently motivated, converged miraculously in the late 1960s, so the word "noticed" in the question has to be placed in context. For example, one needs the notion of topological Steinberg symbol in the study of simple algebraic groups over various local fields. Here an essential contribution was made by Calvin Moore in his 1968 paper Group extensions of p-adic and adelic linear groups (IHES Publ. Math. 35).

The history of all the related developments really ought to be told by one of the living participants. Seminars with Bass in that period got me interested for a while in the subject, leading me eventually to write an elementary introduction in the last part of my 1980 Arithmetic Groups (Springer Lect. Notes 789), where a lot of references are included; most of the key papers are now available online via numdam.org and such, while Steinberg's 1967-68 Yale lectures are posted on some webpages including his at UCLA and Bill Casselman's at UBC. But it would also help to know who was talking to whom during that crucial period.

Briefly, Steinberg's work was purely algebraic at first and was motivated by Chevalley's 1955 Tohoku paper constructing versions of simple adjoint algebraic groups over arbitrary fields. In his 1962 Brussels conference paper, Steinberg worked out generators and relations in order to study non-adjoint groups and in particular see how projective modular representations of the adjoint groups would lift to "universal" groups. This led him to introduce what eventually became Steinberg symbols or cocycles. By the time of his Yale lectures, further connections were in the air. For instance, work on the Congruence Subgroup Problem (Bass, Lazard, Milnor, Serre) led Serre to an elegant formulation of the problem in terms of group extensions. Matsumoto's thesis provided more evidence of the close connection with Steinberg's formalism.

Moore's work on the other hand came from his early interest in locally compact groups and their central extensions. Here he eventually found that a topological version of Steinberg's algebraic formalism would fit well with the classical ideas in number theory (local and global class field theory) explored in modern terms by Serre and others. There is a long and useful review of Moore's paper in Mathematical Reviews by Hyman Bass, if you have access.

However the question in the header is answered, it should be kept in mind that the motivation for arriving at such a bizarre connection came from work of all these people. Connecting the dots required the existence of the dots.

ADDED: A few more remarks about references, influences, timing. 1) Steinberg's 1962 Brussels paper (in French!) is reprinted along with his others in a single volume published by AMS and reasonably priced; but I don't know any accessible online source. However, most of his computations reappeared in his Yale lectures and in Matsumoto's thesis (with special cases treated in my lecture notes). 2) My best guess is that the combined work of Moore and Matsumoto filled in the connection with classical symbols and reciprocity laws. Note that Moore''s paper was submitted in January 1968 but lists Matsumoto's thesis in the references, while Matsumoto's thesis was submitted six months later and cites Moore's published paper. (The year of Moore's paper is either 1968 or 1969, depending on where you look.) Matsumoto thanks Bruhat, Serre, Samuel for their advice. On the other hand, Moore points to independent partial results by T. Kubota. He especially credits his conversations with a number of people including Bass and Serre.

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Thank you very much for your remarks. It is to Steinberg's Brussels paper that I don't have access, but some of it is summarised in his Yale notes. What puzzles me is that everybody in the sixties (Moore, Steinberg, Matsumoto, Serre) seems to treat the Steinberg property of the norm residue symbol as being well known, and I can't find a place where somebody says : Look, what a beautiful property of the norm residue have I discovered ! –  Chandan Singh Dalawat Aug 29 '12 at 12:43
@Chandan: I can update the history a bit, based on some email inquiries I made to participants: Moore was almost certainly the first person to discover the connection between Hilbert symbols and Steinberg cocycles/symbols, no later than early 1966. He gave a colloquium soon afterwards at UCLA with a title like Variations on a theme of Steinberg (with Steinberg in the audience). But everyone else I've mentioned played an essential role in completing the picture, especially the Congruence Subgroup Problem which Moore didn't consider. Very complicated history to reconstruct now. –  Jim Humphreys Sep 13 '12 at 0:36
Yesterday I happened to come across Hasse's Tagebuch which Franz Lemmermeyer and Peter Roquette are editing. Today I noticed the entry 5.1 (dated 3.10.1927) which says that if $k$ is a number field containing a primitive $m$-th root of $1$, and if $\alpha,\beta,\gamma\in k^\times$ satisfy $\alpha+\beta=\gamma$, then $$(\alpha,\beta)_m=(\alpha,\gamma)_m(\gamma,\beta)_m(−1,\gamma)_m$$ at every place of $k$; of course Steinberg's relation is the particular case $\gamma=1$. So Hasse was well aware of this relation in the 20s, and I wonder why he doesn't list it in his Number Theory. –  Chandan Singh Dalawat Sep 24 '12 at 15:58
@Chandan: Yes, the history is indeed complicated. But outside number theory the work of Steinberg (and later Moore) started instead with group extension problems and cocycles. It took a while to connect these apparently unrelated lines of thinking, which Moore seems to have been the first to do. –  Jim Humphreys Sep 30 '12 at 13:35
For the group SL_n this was called the Mennicke symbol in Bass-Milnor-Serre paper and they identify it with hilbert symbol, therefore, maybe Mennicke noticed it . –  Venkataramana Nov 14 '14 at 5:03

Matsumoto's computation of $K_2$ of a field in "Introduction to Algebraic K-Theory" by John Milnor.

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Dear Srilakshmi, I've looked at Matsumoto's article (Annales ENS, 1969). He explicitly says that the norm residue symbol in a local field (other than $\bf C$) is a Steinberg cocycle, but I'm not sure he's the first person to have made that remark. –  Chandan Singh Dalawat Aug 29 '12 at 9:17
Perhaps not. For example on p.87 of Steinberg's 1967 Lectures on Chevalley groups at Yale, he says that these relations [including the Steinberg relation] are satisfied by the norm residue symbol in class field theory. –  Chandan Singh Dalawat Aug 29 '12 at 9:35
Dear Sir, I thought Matsumoto was the first person. –  Srilakshmi Aug 29 '12 at 9:35
Moreover, the first edition of Serre's Corps locaux appeared in 1962, and the Steinberg relation is explicitly mentioned in it without being named as such. –  Chandan Singh Dalawat Aug 29 '12 at 9:53