Effective lower bounds for , the minimal value of the discriminant of algebraic number fields having signature (i.e. having real and non-real conjugates), obtained in 1976 by A.M. Odlyzko. See also Algebraic number; Number field.
The first such bound was proved in 1891 by H. Minkowski [a4], who showed
In 1974, H.M. Stark ([a11], [a12]) observed that Hadamard factorization of the Dedekind zeta-function leads to a formula expressing by the zeros of and the value of its logarithmic derivative at a complex number with . He used this formula with a proper choice of to deduce lower bounds for which were essentially stronger than Minkowski's bound, but did not reach the bounds obtained by geometrical methods.
In particular, one has
If the extended Riemann hypothesis is assumed (cf. also Riemann hypotheses; Zeta-function), then the constants and in (a2) can be replaced by and , respectively. For small degrees the bound (a2) can be improved (see [a2], [a10]) and several exact values of are known.
On the other hand, it has been shown in [a13], as a consequence of their solution of the class field tower problem (cf. also Tower of fields; Class field theory), that is finite. The best explicit upper bound for it, , is due to J. Martinet [a1], who obtained this as a corollary of his constructions of infinite -class towers of suitable fields.
|[a1]||J. Martinet, "Tours de corps de classes et estimations de discriminants" Invent. Math. , 44 (1978) pp. 65–73|
|[a2]||J. Martinet, "Petits discriminants" Ann. Inst. Fourier (Grenoble) , 29 : fasc.1 (1979) pp. 159–170|
|[a3]||J. Martinet, "Petits discriminants des corps de nombres" , Journ. Arithm. 1980 , Cambridge Univ. Press (1982) pp. 151–193|
|[a4]||H. Minkowski, "Théorèmes arithmétiques" C.R. Acad. Sci. Paris , 112 (1891) pp. 209–212 (Gesammelte Abh. I (1911), 261-263, Leipzig–Berlin)|
|[a5]||H.P. Mulholland, "On the product of complex homogeneous linear forms" J. London Math. Soc. , 35 (1960) pp. 241–250|
|[a6]||A. Odlyzko, "Some analytic estimates of class numbers and discriminants" Invent. Math. , 29 (1975) pp. 275–286|
|[a7]||A. Odlyzko, "Lower bounds for discriminants of number fields" Acta Arith. , 29 (1976) pp. 275–297 (II: Tôhoku Math. J., 29 (1977), 275-286)|
|[a8]||A. Odlyzko, "Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: a survey of recent results" Sém. de Théorie des Nombres, Bordeaux , 2 (1990) pp. 119–141|
|[a9]||G. Poitou, "Minoration de discriminants (d'aprés A.M. Odlyzko)" , Sém. Bourbaki (1975/76) , Lecture Notes in Mathematics , 567 , Springer (1977) pp. 136–153|
|[a10]||G. Poitou, "Sur les petits discriminants" Sém. Delange–Pisot–Poitou , 18 : 6 (1976/77)|
|[a11]||H.M. Stark, "Some effective cases of the Brauer–Siegel theorem" Invent. Math. , 23 (1974) pp. 135–152|
|[a12]||H.M. Stark, "The analytic theory of numbers" Bull. Amer. Math. Soc. , 81 (1975) pp. 961–972,|
|[a13]||E.S. Golod, I.R. Shafarevich, "On the class-field tower" Izv. Akad. Nauk. SSSR , 28 (1964) pp. 261–272 (In Russian)|
Odlyzko bounds. WÅ‚adysÅ‚aw Narkiewicz (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Odlyzko_bounds&oldid=15182