The arithmetic lifting property for nilpotent groups [Elektronische Ressource] / vorgelegt von Sebastian Basten

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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.

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Published 01 January 2011
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INAUGURAL – DISSERTATION
zurErlangungderDoktorwu¨rdeder Naturwissenschaftlich-MathematischenGesamtfakulta¨t derRuprecht-Karls-Universit¨atHeidelberg
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 extensionMKwith 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 GaloiserweiterungMKmit 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
Contents Introduction 7 A note on notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1 Prerequisites 13 1.1 Specialization and valuations . . . . . . . . . . . . . . . . . . . . . . . 13 1.2 Some basic invariant theory . . . . . . . . . . . . . . . . . . . . . . . 15 1.3 The arithmetic lifting property . . . . . . . . . . . . . . . . . . . . . 17 1.3.1 The Noether problem and generic polynomials . . . . . . . . . 17 1.3.2 The arithmetic lifting property . . . . . . . . . . . . . . . . . 17 1.3.3 The arithmetic lifting property for nilpotent groups . . . . . . 19 2p-groups and embedding problems 21 2.1p-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.1.1 Nilpotent groups . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.1.2 Commutators and collection . . . . . . . . . . . . . . . . . . . 22 2.2 Embedding problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.2.1 Basic theory of embedding problems . . . . . . . . . . . . . . 26 2.2.2 Brauer type embedding problems . . . . . . . . . . . . . . . . 28 2.2.3 Brauer type embedding problems and cohomology . . . . . . . 29 2.2.4 Factor systems . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.3 Embedding problems forp-groups . . . . . . . . . . . . . . . . . . . . 37 3 The arithmetic lifting property for nilpotent groups 39 3.1 The arithmetic lifting property for factor groups . . . . . . . . . . . . 39 3.2 The arithmetic lifting property for embedding problems . . . . . . . . 40 3.3 The arithmetic lifting property for nilpotent groups overQab. . . . . 67 3.4 Outlook: The arithmetic lifting property for solvable groups . . . . . 69 4 Some other results on the arithmetic lifting property 71 4.1 Other results on the arithmetic lifting property . . . . . . . . . . . . 71 4.2 Lists of groups having the arithmetic lifting property . . . . . . . . . 72 4.2.1 List of groups having the arithmetic lifting property . . . . . . 72 4.2.2 List of groups with generic polynomials . . . . . . . . . . . . . 73 5
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Introduction
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 extensionsMK0of all fieldsK0Kcan be obtained by specializing thetito valuesaiK0and 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-extensionMKis 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 ¯ ˜ ¯ KM=Kfor every algebraic closureKofK. 7