The Second Chinburg Conjecture for Quaternion Fields

Abstract

<p>This 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.</p>