Infinite Global Fields and the Generalized Brauer-Siegel Theorem

The paper has two purposes. First, we start to develop a theory of infinite global fields, i.e., of infinite algebraic extensions either of Q or of Fr(t). We produce a series of invariants of such fields, and we introduce and study a kind of zeta-function for them. Second, for sequences of number fields with growing discriminant we prove generalizations of the Odlyzko-Serre bounds and of the Brauer-Siegel theorem, taking into account non-archimedean places. This leads to asymptotic bounds on the ratio log hR/log p |D| valid without the standard assumption n/log p |D| → 0, thus including, in particular, the case of unramified towers. Then we produce examples of class field towers, showing that this assumption is indeed necessary for the Brauer-Siegel theorem to hold. As an easy consequence we ameliorate on existing bounds for regulators.

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