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• W2082117283 abstract "Let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper K slash bold upper Q> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=bold>Q</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>K/mathbf {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an abelian extension and let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper D> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding=application/x-tex>D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the absolute value of the discriminant of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper K> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding=application/x-tex>K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We show that for each <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=epsilon greater-than 0> <mml:semantics> <mml:mrow> <mml:mi>ε<!-- ε --></mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>varepsilon > 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the smallest rational prime that splits completely in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper K> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding=application/x-tex>K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper O left-parenthesis upper D Superscript one fourth plus epsilon Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:msup> <mml:mi>D</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>4</mml:mn> </mml:mfrac> <mml:mo>+</mml:mo> <mml:mi>ε<!-- ε --></mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>O(D^{frac 14+varepsilon })</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Here the implied constant depends only on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=epsilon> <mml:semantics> <mml:mi>ε<!-- ε --></mml:mi> <mml:annotation encoding=application/x-tex>varepsilon</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the degree of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper K> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding=application/x-tex>K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This generalizes a theorem of Elliott, who treated the case when <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper K slash bold upper Q> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=bold>Q</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>K/mathbf {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has prime conductor." @default.
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• W2082117283 date "2014-03-05" @default.
• W2082117283 modified "2023-09-28" @default.
• W2082117283 title "The smallest prime that splits completely in an abelian number field" @default.
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