Hilbert Symbols (History)

Question

Usually the Hilbert symbol $\left(\frac{a,b}{p}\right)$ is defined via the solvability of $z^2=ax^2+by^2$ in the $p$-adic numbers. Obviously, this can not have been Hilberts original definition, because $p$-adic numbers were only invented later. (And I guess solubility in the $p$-adics is not the same as solubility modulo $p$, right?) What was the original definition?

Answer 1

See Hilbert symbol :

The Hilbert symbol was introduced by David Hilbert in his Zahlbericht (1897), with the slight difference that he defined it for elements of global fields rather than for the larger local fields.

For English translation, see David Hilbert, The Theory of Algebraic Number Fields (1897), §64 The Symbol $(\dfrac {n,m}{w})$, page 120.

Answer 2

First, I don't think that your chronological objection (concerning $\mathbf Q_p$, Hensel and Hilbert) is founded: 1897 was the year of birth of the $p$-adics, but also the year of publication of Hilbert's Zahlbericht. As the preeminent mathematician of his time, Hilbert had certainly been aware of Hensel's work.

Second, in spite of not being an expert in the history of number theory, I feel reasonably assured that the origin of the Hilbert symbol must be traced back to the Legendre symbol and Gauss' quadratic reciprocity law. Recall that, for a prime $p$ not dividing $d$, by definition $(\frac d p) =+1$ if $d$ is a quadratic residue mod $p$, $-1$ if not, and for two distinct prime numbers $p$ and $q$, $(\frac p q)(\frac q p)$=$(-1)^{ (p-1)(q-1)/4}$. The arguments $d,p$ are in $\mathbf Z$ and the value of the symbol decides whether $d$ is a square in $\mathbf F_p^*$. This is the "residual part" of the theory.

The "local part" introduces the Hilbert symbol $(a,b)_p$ which you cite, with arguments $a,b \in\mathbf Q_p^*$. By definition $(a,b)_p = +1$ if the quadratic form $z^2-ax^2-by^2$ represents $0$ in $\mathbf Q_p$ (i.e. admits a non zero isotropic vector), $-1$ if not. The point is that the Hilbert symbol can be explicitly given in terms of Legendre symbols. The most important application is a classification of finite dimensional non degenerate quadratic forms over $\mathbf Q_p$. If such a form is written in diagonal form, with coefficients $a_1,..., a_n$, it is characterized up to equivalence by three invariants: its dimension $n$; its discriminant $\delta:=\prod a_i$ (viewed in $\mathbf Q_p^*/{\mathbf Q_p^*}^2$ ); a third invariant defined as $\epsilon:=\prod (a_i,a_j)_p$, for all pairs $i<j$.

In the "passage from local to global", it is absolutely necessary to introduce $\mathbf R$, sometimes denoted $\mathbf Q_\infty$, which must be set on the same footing as all the $\mathbf Q_p$'s, with a Hilbert symbol $(a,b)_\infty$ defined analogously. The generic notation for $p$ and $\infty$ is $v$, called a "place" of $\mathbf Q$, and the "product formula" $\prod (a,b)_v=1$ (where $v$ runs over all the places of $\mathbf Q$) is the generalization of the quadratic reciprocity law. The most important result about quadratic forms over $\mathbf Q$ is the Hasse-Minkowski theorem: Let $f$ be a non degenerate quadratic form with coefficients in $\mathbf Q$. Then $f$ represents $0$ in $\mathbf Q$ iff, for all places $v$, $f_v$ (obtained by considering the coefficients as living in $\mathbf Q_v$) represents $0$ in $\mathbf Q_v$. For all these results see e.g. Serre's "A course in Arithmetic", chap. 3-4.

NB: $(\pm 1)$ is no other than the group $\mu_2$ of square roots of unity. For a number field $K$ containing the group $\mu_n$ of $n$-th roots of unity, one can define local Hilbert symbols $(.,.)_v$ taking values in $\mu_n$. This is the starting point of the so-called "explicit reciprocity laws" in class-field theory. See e.g. Cassels-Fröhlich, "Algebraic Number Theory", exercises 1-2 of the appendix (not elementary).