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L-series and homomorphisms of number fields

While the zeta function does not determine a number field uniquely, the $L$-series of a well-chosen Dirichlet character does. Moreover, isomorphisms between two number fields are in natural bijection with $L$-series preserving isomorphisms of $l$-torsion subgroups of the Dirichlet character groups. We extend this by showing that homomorphisms between number fields are in natural bijection with group homomorphisms between $l$-torsion subgroups of the Dirichlet character groups abiding a divisibility condition on the $L$-series when $l$ is sufficiently large.