Abstract
In this chapter we shall examine invariants of the Galois module structure on the higher-dimensional algebraic K-groups of rings of algebraic integers in number fields. Special cases of these invariants were given in §§1.3.4-1.3.13. The material of this chapter originated in ([132] Chapter VII) where generalizations of the Chin-burg invariant, Ω0(L/K, 3) of [34] were constructed using algebraic K-groups of integers in dimensions two and three. Independently, Pappas (unpublished) had noticed the same construction. In [132] it was shown, in the totally real case, that the image of this new invariant in the kernel group, D(Z[G(L/K)]) of §2.5.26, is represented in the Horn-description of Theorem 2.5.32 by the value of the Artin L-function at s = -1. The examples described in §§1.3.4-1.3.13 provide further evidence to connect the invariant, Ωr (L/K, 3) constructed from K-groups in dimesions 2r and 2r + 1, with behaviour of the Artin L-series at (or near) s = -r.
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© 2002 Springer Basel AG
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Snaith, V.P. (2002). Higher K-theory of Algebraic Integers. In: Algebraic K-Groups as Galois Modules. Progress in Mathematics, vol 206. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8207-1_5
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DOI: https://doi.org/10.1007/978-3-0348-8207-1_5
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9473-9
Online ISBN: 978-3-0348-8207-1
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