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On a theorem of Mislin
No full textAn alternative proof is given of a result, originally due to Guido Mislin, giving necessary and sufficient conditions for the inclusion of a subgroup to induce an isomorphism in mod p cohomolog
Hecke algebras and class-groups of integral group-rings
No full textLet G be a finite group. To a set of subgroups of order two we associate a mod 2 Hecke algebra and construct a homomorphism, ?, from its units to the class-group of Z[G]. We show that this homomorphism takes values in the subgroup, D(Z[G]). Alternative constructions of Chinburg invariants arising from the Galois module structure of higher-dimensional algebraic K-groups of rings of algebraic integers often differ by elements in the image of ?. As an application we show that two such constructions coincide
Comparison of K-theory Galois module structure invariants
No full textWe prove that two, apparently different, class-group valued Galois module structure invariants associated to the algebraic -groups of rings of algebraic integers coincide. This comparison result is particularly important in making explicit calculations
The Second Chinburg Conjecture for Quaternion Fields
No full textThis thesis is a part of a program to study the Second Chinburg Conjecture. Let N be a quaternion extension of the rational; containing Q(√d₁,√d₂), where d₁ ≡ 3 (mod 8) and d₂ ≡ 10 (mod 16). A projective Z[Q₈]-module inside the ring of integers ON is constructed and is used, together with a cohomological classification of cohomologically trivial, 2-primary Q-modules, to compare Ω(N/Q,2), Chinburg's second invariant, with WN/Q, the root number class defined by Ph. Cassou-Nougès and A. Fröhlich. The Second Chinburg Conjecture for this extension N/Q is confirmed. Together with results of J. Hooper and S. Kim this calculation verifies the Second Chinburg Conjecture for all quaternion extensions of the rationals.Doctor of Philosophy (PhD
Some Families of Quaternion Fields and the Second Chinburg Conjecture
No full textLet N/K be a finite Galois extension of number fields with Galois group G. The second Chinburg conjecture asserts the equivalence in the locally free class group CL(Z[G]) of the classes corresponding to two arithmetic invariants attached to the extension, namely the Frōhlich-Cassou-Noguès class Wɴ/k and the second Chinburg invariant Ω(ɴ/k,2). Frōhlich originally formulated a conjectural equivalence of the class Wɴ/k and the class [Oɴ] determined by the ring of integers in N, and this was subsequently verified by M. Taylor. The second Chinburg invariant Ω(N/K,2) may be interpreted as a generalisation of the class [Oɴ] to wild extensions, and Chinburg showed that Frōhlich's conjecture for tame extensions implied Chinburg's conjecture for tame extensions. More generally, the second Chinburg conjecture has been verified when G is absolutely abelian of odd conductor (C. Greither) and for several infinite families of wildly ramified quaternion fields (S. Kim). When N/K is wildly ramified and G is nonabelian, little further evidence for the conjecture is known. In this thesis we consider some families of extensions N/Q with G ≅ Q₈, the quaternion group of order 8, in which the prime 2 is totally ramified. In particular we consider those quatermon extensions containing a biquadratic subfield of the form Q(√a, √b), where a ≡ 2 mod 16 and b ≡ mod 8. We verify the second Chinburg conjecture for all such extensions. We begin by localizing at the prime 2 and make use of a cohomological classification of a large class of cohomologically trivial 2-primary Z[Q₈]-modules to explicitly compute the local second Chinburg invariant. Globally, we construct a projective Z[Q₈]-module, X, that lies inside ON, the ring of integers of N. We then work with Frōhlich's Hom-description of the class group and show that, in fact, [X] = Wɴ/Q in CL(Z[Q]), where [X] denotes the class of X. Combining this with the local information, and using congruence methods, we conclude that in CL(Z[Q₈]), Ω(ɴ/Q,2) = Wɴ/Q, i.e., the second Chinburg conjecture holds for these extensions. Combining this result with work of S. Kim, V. Snaith and M. Tran, this establishes the second Chinburg conjecture for all extensions ɴ/Q having Galois group G ≅ Q₈.Doctor of Philosophy (PhD
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
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