Small generators of number fields |
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Authors: | Wolfgang M Ruppert |
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Institution: | Mathematisches Institut, Universit?t Erlangen-Nürnberg, Bismarckstra?e 1?, D-91054 Erlangen, Germany.?e-mail: ruppert@mi.uni-erlangen.de, DE
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Abstract: | If K is a number field of degree n over Q with discriminant D
K
and if α∈K generates K, i.e. K=Q(α), then the height of α satisfies with . The paper deals with the existence of small generators of number fields in this sense. We show: (1) For each $n$ there are
infinitely many number fields K of degree $n$ with a generator α such that . (2) There is a constant d
2 such that every imaginary quadratic number field has a generator α which satisfies .?(3) If K is a totally real
number field of prime degree n then one can find an integral generator α with .
Received: 10 January 1997 / Revised version: 13 January 1998 |
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Keywords: | Mathematics Subject Classification (1991):11Y40 11Y16 11R99 |
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