Did Hermite really prove "Hermite's Theorem" on number field discriminants? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T10:39:54Z http://mathoverflow.net/feeds/question/103148 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/103148/did-hermite-really-prove-hermites-theorem-on-number-field-discriminants Did Hermite really prove "Hermite's Theorem" on number field discriminants? Frank Thorne 2012-07-26T02:39:31Z 2013-04-23T20:08:22Z <p><em>Hermite's theorem</em>, as it is typically called, is that there are only finitely many number fields of bounded (equivalently, fixed) discriminant.</p> <p>The usual proof (see Neukirch's <em>Algebraic Number Theory</em> for example) proceeds as follows. First, one proves the <em>Minkowski bound</em>: $|Disc(K)| > (C + o(1))^{deg \ K}$ where $C > 1$. This reduces the problem to fields of fixed degree. For fields $K$ of degree $n$ with $Disc(K) &lt; X$, one can then obtain bounds on the coefficients of the minimal polynomial of $K$, and in particular there are only finitely many possibilities.</p> <p>However, Minkowski's work was more than thirty years after Hermite's. I looked at <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN243919689_0053&amp;DMDID=DMDLOG_0014" rel="nofollow">Hermite's original paper</a> 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. </p> <p>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?</p> <p>And regardless of whether he actually did, might he have reasonably done so other than coming up with Minkowski's bound on his own?</p> <p>Thank you!</p> http://mathoverflow.net/questions/103148/did-hermite-really-prove-hermites-theorem-on-number-field-discriminants/128521#128521 Answer by Rhett Butler for Did Hermite really prove "Hermite's Theorem" on number field discriminants? Rhett Butler 2013-04-23T18:47:23Z 2013-04-23T18:52:39Z <p>Hermite and Minkowski used completely different ideas.</p> <p>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.</p> <p>Minkowsky, 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: <em>Minkowski's proof of a deep number theoretic theorem without computational help, mainly on geometrical vizualization, is a pearl of Minkowski's inventiveness</em>.</p> <p>A masterly and/but very readable presentation of both ideas has been given by <a href="http://www.math.tifr.res.in/~publ/ln/tifr07.pdf" rel="nofollow">C.L. Siegel: Lectures on Quadratic Forms</a></p> http://mathoverflow.net/questions/103148/did-hermite-really-prove-hermites-theorem-on-number-field-discriminants/128531#128531 Answer by Yazdegerd III for Did Hermite really prove "Hermite's Theorem" on number field discriminants? Yazdegerd III 2013-04-23T20:08:22Z 2013-04-23T20:08:22Z <p>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).</p>