Abstract
We prove that two global fields are isomorphic if and only if there is an isomorphism of groups of Dirichlet characters that preserves L-series.
Highlights
As was discovered by Gaßmann in 1926 [10], number fields are not uniquely determined up to isomorphism by their zeta functions
We prove that two global fields K and L are isomorphic if and only if there exists an isomorphism of groups of Dirichlet characters ψq : GqKab −→ ∼ GqLab that preserves L-series: LK(χ ) = LL(ψq (χ )) for all χ ∈ GqKab
A more detailed series of equivalences can be found in the Main Theorem 3.1 below. To connect this theorem to the discussion in the previous paragraph, observe that the existence of ψq without equality of L-series is the same as K and L having abelianized Galois groups that are isomorphic as topological groups, and that for the trivial character χtriv, we have LK(χtriv) = ζK, so that preserving L-series at χtriv is the same as K and L having the same zeta function
Summary
As was discovered by Gaßmann in 1926 [10], number fields are not uniquely determined up to isomorphism by their zeta functions. We prove that two global fields K and L are isomorphic if and only if there exists an isomorphism of groups of Dirichlet characters ψq : GqKab −→ ∼ GqLab that preserves L-series: LK(χ ) = LL(ψq (χ )) for all χ ∈ GqKab. A more detailed series of equivalences can be found in the Main Theorem 3.1 below. A more detailed series of equivalences can be found in the Main Theorem 3.1 below To connect this theorem to the discussion in the previous paragraph, observe that the existence of ψq without equality of L-series is the same as K and L having abelianized Galois groups that are isomorphic as topological groups, and that for the trivial character χtriv, we have LK(χtriv) = ζK, so that preserving L-series at χtriv is the same as K and L having the same zeta function. We do not need the full hypothesis: we can prove the stronger result that for every number field, there exists a character of any chosen order > 2 for which the L-series does not equal any other Dirichlet L-series of any other field (see Theorem 10.1)
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