INAUGURAL { DISSERTATIONzur Erlangung der Doktorwurde derNaturwissenschaftlich-Mathematischen Gesamtfakult atder Ruprecht-Karls-Universit at Heidelbergvorgelegt vonDipl.-Math. Sebastian Bastenaus Frankfurt am MainTag der mundlic hen Prufung: 14.04.2011ThemaThe arithmetic lifting property fornilpotent groupsGutachter: Prof. Dr. Bernd Heinrich MatzatProf. Dr. Michael DettweilerAbstractIn this thesis it is shown that every nite nilpotent group has the arithmetic liftingabproperty overQ , the maximal abelian extension of the eld of rational numbers.For a group G to have the arithmetic lifting property over a eld K means thatevery Galois extension M=K with Galois group G can be obtained from a Galois~extension M=K(t), regular over K, with Galois group G by replacing the variablet with an element of K. In particular it is shown that every nite nilpotent groupabcan be realized regularly as Galois group overQ (t).ZusammenfassungIn dieser Arbeit wird gezeigt, dass jede endliche nilpotente Gruppe die Arithmetis-abche Liftungseigenschaft ub erQ hat, der maximalen abelschen Erweiterungen desK orpers der rationalen Zahlen.Hierbei hat eine Gruppe G die Arithmetische Liftungseigenschaft ub er eine Korp erK, wenn jede Galoiserweiterung M=K mit Galoisgruppe G aus einer ub er K regu-~arenl Erweiterung M=K(t) gewonnen werden kann, indem die Variable t durch einElement aus K ersetzt wird.
vorgelegt von Dipl.-Math. Sebastian Basten aus Frankfurt am Main TagdermundlichenPru¨fung:14.04.2011 ¨
The
Thema
arithmetic lifting property nilpotent groups
Gutachter:
for
Prof. Dr. Bernd Heinrich Matzat Prof. Dr. Michael Dettweiler
Abstract In this thesis it is shown that every finite nilpotent group has the arithmetic lifting property overQab, the maximal abelian extension of the field of rational numbers. For a groupGto have the arithmetic lifting property over a fieldKmeans that every Galois extensionMKwith Galois groupGcan be obtained from a Galois ˜ extensionM K(t), regular overK, with Galois groupGby replacing the variable twith an element ofK. In particular it is shown that every finite nilpotent group can be realized regularly as Galois group overQab(t).
Zusammenfassung In dieser Arbeit wird gezeigt, dass jede endliche nilpotente Gruppe die Arithmetis-cheLiftungseigenschaftu¨berQabhat, der maximalen abelschen Erweiterungen des Ko¨rpersderrationalenZahlen. Hierbei hat eine GruppeGeAriditfnuehiLitcshtemubt¨afchnsgeeigsrepro¨Keniere K, wenn jede GaloiserweiterungMKmit GaloisgruppeGaiesur¨neerubKregu-˜ la¨renErweiterungM K(t) gewonnen werden kann, indem die Variabletdurch ein Element ausK wird nachgewiesen, dass jede endliche Insbesondereersetzt wird. nilpotente Gruppe regul¨ ¨berQab(t) realisiert werden kann. ar u
The originalNoether problemformulated by E. Noether in [Noether1] poses the question, whether the fixed field of a permutation groupGwhich acts on a rational function field by permuting the indeterminates is again purely transcendental over the base field. When this is the case, it is possible to obtain a parameterization for all polynomials with Galois groupG. Even very small groups do not necessarily have this property: The first counterexamples over the field of rational numbersQ are abelian groups which contain at least one element of order 8 by [Lenstra1] and C47cyclic group of order 47, by [Swan1]., the A weaker property of a groupGis the existence ofgeneric polynomialsfor this group over a given fieldK. A generic polynomial is a polynomialg(t1, ..., tn, X) which has Galois groupGover the rational function fieldK(t1, .., tn) innindetermi-nates such that all Galois extensionsMK0of all fieldsK0⊇Kcan be obtained by specializing thetito valuesai∈K0and taking the splitting field ofg(a1, ..., an, X). For infinite fields the existence of generic polynomials for a given group is equivalent to the existence ofgeneric extensions Generic polyno-as described in [Saltman1]. mials overQcyclic groups of odd order, thus for someexist, for example, for all groups which do not have the properties considered in the Noether problem. For abelian groups which contain elements of order 8, generic polynomials still do not exist by [Saltman1]. If there are no generic polynomials for a given group over a given field, one can ask if there is a “weaker specialization property” which is satisfied by this group. Thearithmetic lifting property, formulated for the first time in [Beckmann1] in 1992, is such a property. By definition a finite groupGhas thearithmetic lifting property over a given fieldKif everyG-extensionMKis a specialization of aG-extension ˜ M K(t) ofK(tone variable, that is regular over), the rational function field in K. ˆ M G{{{{@@@ {@@ {{{@@@ K(t)M } E EEEEEE}}G}}}}} E K ˜ In this context, a field extensionM K(t) is calledregular (over K)orgeometric, if ¯ ˜ ¯ K∩M=Kfor every algebraic closureKofK. 7