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W1966321013 startingPage "385" @default.
W1966321013 abstract "Let <italic>l</italic> be a prime number and <italic>m</italic> a divisor of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=l minus 1> <mml:semantics> <mml:mrow> <mml:mi>l</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>l - 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Then the Gauss period <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=omega equals zeta plus zeta Superscript lamda Baseline plus zeta Superscript lamda squared Baseline plus midline-horizontal-ellipsis plus zeta Superscript lamda Super Superscript m minus 1> <mml:semantics> <mml:mrow> <mml:mi>ω</mml:mi> <mml:mo>=</mml:mo> <mml:mi>ζ</mml:mi> <mml:mo>+</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>ζ</mml:mi> <mml:mi>λ</mml:mi> </mml:msup> </mml:mrow> <mml:mo>+</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>ζ</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>λ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:mrow> </mml:msup> </mml:mrow> <mml:mo>+</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>+</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>ζ</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>λ</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>m</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:mrow> </mml:msup> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>omega = zeta + {zeta ^lambda } + {zeta ^{{lambda ^2}}} + cdots + {zeta ^{{lambda ^{m - 1}}}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=zeta equals e Superscript 2 pi i slash l> <mml:semantics> <mml:mrow> <mml:mi>ζ</mml:mi> <mml:mo>=</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>2</mml:mn> <mml:mi>π</mml:mi> <mml:mi>i</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>l</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>zeta = {e^{2pi i/l}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=lamda> <mml:semantics> <mml:mi>λ</mml:mi> <mml:annotation encoding=application/x-tex>lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a primitive <italic>m</italic>th root of unity modulo <italic>l</italic>, generates a subfield <italic>K</italic> of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper Q left-parenthesis zeta right-parenthesis> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>Q</mml:mi> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>ζ</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbb {Q}(zeta )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of degree <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis l minus 1 right-parenthesis slash m> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>l</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=false>)</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>m</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>(l - 1)/m</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In this paper we study the reciprocal minimal polynomial <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper F Subscript l comma m Baseline left-parenthesis upper X right-parenthesis equals upper N Subscript upper K slash double-struck upper Q Baseline left-parenthesis 1 minus omega upper X right-parenthesis> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>F</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>l</mml:mi> <mml:mo>,</mml:mo> <mml:mi>m</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>N</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <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=double-struck>Q</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:mi>ω</mml:mi> <mml:mi>X</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>{F_{l,m}}(X) = {N_{K/mathbb {Q}}}(1 - omega X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=omega> <mml:semantics> <mml:mi>ω</mml:mi> <mml:annotation encoding=application/x-tex>omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> over <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper Q> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>Q</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbb {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. It will be shown that for fixed <italic>m</italic> and every <italic>N</italic> we have <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper F Subscript l comma m Baseline left-parenthesis upper X right-parenthesis identical-to left-parenthesis upper B Subscript m Baseline left-parenthesis upper X right-parenthesis Superscript l Baseline slash left-parenthesis 1 equals m upper X right-parenthesis right-parenthesis Superscript 1 slash m Baseline left-parenthesis mod upper X Superscript upper N Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>F</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>l</mml:mi> <mml:mo>,</mml:mo> <mml:mi>m</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>≡</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>m</mml:mi> </mml:msub> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>(</mml:mo> <mml:mi>X</mml:mi> <mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mi>l</mml:mi> </mml:msup> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>=</mml:mo> <mml:mi>m</mml:mi> <mml:mi>X</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>1</mml:mn> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>m</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:mspace width=thickmathspace /> <mml:mspace width=0.667em /> <mml:mo stretchy=false>(</mml:mo> <mml:mi>mod</mml:mi> <mml:mspace width=0.333em /> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>X</mml:mi> <mml:mi>N</mml:mi> </mml:msup> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>{F_{l,m}}(X) equiv {({B_m}{(X)^l}/(1 = mX))^{1/m}};pmod {{X^N}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for all but finitely many exceptional primes <italic>l</italic> (depending on <italic>m</italic> and <italic>N</italic>), where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper B Subscript m Baseline left-parenthesis upper X right-parenthesis element-of double-struck upper Z left-bracket left-bracket upper X right-bracket right-bracket> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>m</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>∈</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>Z</mml:mi> </mml:mrow> <mml:mo stretchy=false>[</mml:mo> <mml:mo stretchy=false>[</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy=false>]</mml:mo> <mml:mo stretchy=false>]</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>{B_m}(X) in mathbb {Z}[[X]]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a power series depending only on <italic>m</italic> . A method of computation of this set of exceptional primes is presented. The generalization of the results to the case of composite <italic>l</italic> is also discussed." @default.
W1966321013 created "2016-06-24" @default.
W1966321013 creator A5005589416 @default.
W1966321013 creator A5019292615 @default.
W1966321013 date "1993-01-01" @default.
W1966321013 modified "2023-09-27" @default.
W1966321013 title "On the coefficients of the minimal polynomials of Gaussian periods" @default.
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W1966321013 doi "https://doi.org/10.1090/s0025-5718-1993-1155574-7" @default.
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