Small generators of abelian number fields

We show that for each abelian number field K of sufficiently large degree d there exists an element <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>α</m:mi> <m:mo>∈</m:mo> <m:mi>K</m:mi> </m:mrow> </m:math> {α∈ K} with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>K</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mi>ℚ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>α</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {K=Q(α)} and absolute Weil height <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>H</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>α</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:msub> <m:mo>≪</m:mo> <m:mi>d</m:mi> </m:msub> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:msub> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>K</m:mi> </m:msub> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mfrac> <m:mn>1</m:mn> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>d</m:mi> </m:mrow> </m:mfrac> </m:msup> </m:mrow> </m:math> {H(α)d|ΔK|1/2d} , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>K</m:mi> </m:msub> </m:math> {ΔK} denotes the discriminant of K . This answers a question of Ruppert from 1998 in the case of abelian extensions of sufficiently large degree. We also show that the exponent <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mfrac> <m:mn>1</m:mn> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>d</m:mi> </m:mrow> </m:mfrac> </m:math> {1/2d} is best-possible when d is even.