We begin with the historical background to the second Chinburg conjecture.
If L/K is a Galois extension of number fields with Galois group, G G(L/K),
the Normal Basis Theorem states that L is a free if [G]-module of rank one. By
analogy, if N/Q is a Galois extension of number fields with group, G = G(N/Q),
one may ask: Under what conditions is ON, the ring of integers in JV, a free
Z[G]-module? In order to be free, it is necessary that ON be locally free, which
happens precisely when N/Q is tamely ramified, by Noether's Theorem (c.f. [19]
pp.26-27). In fact, when N/Q is tamely ramified and abelian, Hilbert showed the
existence of a normal integral basis (c.f. [19]). In 1969, J. Martinet [35] proved
the existence of a normal integral basis for all tame extensions N/Q having Galois
group G(N/Q) = D2P, the dihedral group of order 2p, where p is an odd prime.
On the other hand, he discovered tame quaternion extensions, N/Q, for which
no normal integral basis exists. Calculations of A. Frohlich, J. Martinet and J-
P. Serre suggested that the existence of a normal integral basis for such tamely
ramified quaternion fields was determined by the sign of the Artin root number of
the (unique) irreducible symplectic representation of the Galois group.
Frohlich was led, by this evidence, to conjecture that the class of ON in the
class-group of Z[G] was equal to the root number class, WN/K, whose definition is
given in §1.10. The root number class is constructed, via the Hom-description of
Theorem 1.4, from the Artin root numbers of the symplectic representations of the
Galois group.
This conjecture was settled by M.J. Taylor:
Theorem ([54])
Let N/K be a tame Galois extension of number fields with G G(N/K). Then
Although the ring of integers, ON, does not define a class in the class-group
of Theorem 1.4 when the extension is wildly ramified, the Cassou-Nogues-Frohlich
is always defined. Therefore the question arises of how to generalise
the class, [ON], in the class-group. This generalisation is given by the second
Chinburg invariant, Ct(L/K,2), of ([10],[9]). This invariant is associated to any
Galois extension of number fields and its definition is given in §§3.3-3.4. In view of
the above theorem the following conjecture seems quite reasonable:
The Second Chinburg Conjecture ([7], [8], [9]))
Let L/K be a finite Galois extension of number fields. Then
Sl(L/K,2) = WL/K e CC(Z[G(L/K)])
where WL/K is as in §1.10.
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