Did Hermite really prove "Hermite's Theorem" on number field discriminants?

Question

Hermite's theorem, as it is typically called, is that there are only finitely many number fields of bounded (equivalently, fixed) discriminant.

The usual proof (see Neukirch's Algebraic Number Theory for example) proceeds as follows. First, one proves the Minkowski bound: $|Disc(K)| > (C + o(1))^{deg \ K}$ where $C > 1$. This reduces the problem to fields of fixed degree. For fields $K$ of degree $n$ with $Disc(K) < X$, one can then obtain bounds on the coefficients of the minimal polynomial of $K$, and in particular there are only finitely many possibilities.

However, Minkowski's work was more than thirty years after Hermite's. I looked at Hermite's original paper and although I confess to not having read it in detail, it seems to be essentially the proof I described above. In particular he only claims on the first page to prove the theorem for fields of a fixed degree, and there is nothing I found in the paper which looks like it applies to all degrees.

So did Hermite actually prove the result that bears his name, or has he been given credit for the jazzed-up version, which apparently could only have been proved thirty years later?

And regardless of whether he actually did, might he have reasonably done so other than coming up with Minkowski's bound on his own?

Thank you!

Answer 1

Hermite and Minkowski used completely different ideas.

Hermite, by induction on $n$, proved that a positive definite quadratic form $q = X^TAX$ of determinant $ |A| = 1$ and $n$-dimensional column-vector $X$ takes a nonzero value $q \leq (4/3)^{(n-1)/2}$ for some integer vector.

Minkowski, by a purely geometrical consideration, discovering that spheres of radius of half the distance of two nearest lattice-points could be replaced by any symmetric convex body, achieved his estimation which is better than Hermite's for large $n$. Hilbert, in his obituary, praised: Minkowski's proof of a deep number theoretic theorem without computational help, mainly on geometrical vizualization, is a pearl of Minkowski's inventiveness.

A masterly and/but very readable presentation of both ideas has been given by C.L. Siegel: Lectures on Quadratic Forms

Answer 2

What Rhett Bulter says is also detailed in a chapter entitled 'From Hermite to Minkowski' in a book of Scharlau and Opolka (From Fermat to Minkowski, Lectures on the Theory of Numbers and Its Historical Development, Springer, 1984).

Answer 3

Indeed Hermite did not prove what today usually is called Hermite's theorem. Translated into modern terms, he shows that there are finitely many number fields of given degree and given discriminant. Since this was way before Dedekind introduced integral bases and discriminants of number fields, he had to talk about roots of monic polynomials with integral coefficients and discriminants of polynomials. The fact that the degree was bounded is due to Minkowski, as you suspected. Even if Hermite had known the notion of discriminants of number fields I cannot see how to prove that the degree is bounded with his methods.