Bounds for small generators of number fields
Let k be an algebraic number field with degree d over the rational field Q, and
discriminant ∆k . If k has a real embedding then we prove that k has a generator α
such that H(α) ≤ |∆k |1/2d , where H(α) is the absolute multiplicative Weil height.
This verifies a conjecture of W. Ruppert. If k has no real embedding the situation is
more complicated, and we are only able to obtain a conditional result. In this case
we prove a similar bound on the height of a generator for a number fields k, but we
must assume that the Dedekind zeta-function associated to the Galois closure of
k/Q satisfies the generalized Riemann hypothesis. This is joint work with Martin
Widmer.
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