Let k be a given positive odd integer and p an odd prime. In this paper, we shall give a sufficient condition for when a prime p divides the order of the groups <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1177_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="123" /> </InlineMediaObject> <EquationSource Format="TEX">$K_{2k}(\mathbb {Z}[\zeta _m+\zeta _m^{-1}])$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>K</mi> <mrow> <mn>2</mn> <mi>k</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Z</mi> <mrow> <mo stretchy="false">[</mo> <msub> <mi>ζ</mi> <mi>m</mi> </msub> <mo>+</mo> <msubsup> <mi>ζ</mi> <mi>m</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">]</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1177_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="79" /> </InlineMediaObject> <EquationSource Format="TEX">$K_{2k}(\mathbb {Z}[\zeta _m])$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>K</mi> <mrow> <mn>2</mn> <mi>k</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Z</mi> <mrow> <mo stretchy="false">[</mo> <msub> <mi>ζ</mi> <mi>m</mi> </msub> <mo stretchy="false">]</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1177_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$\zeta _m$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ζ</mi> <mi>m</mi> </msub> </math></EquationSource> </InlineEquation> is a primitive mth root of unity. When F is a p-extension contained in <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1177_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="37" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {Q}(\zeta _\ell )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">Q</mi> <mo stretchy="false">(</mo> <msub> <mi>ζ</mi> <mi>ℓ</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for some prime <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1177_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\ell $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℓ</mi> </math></EquationSource> </InlineEquation>, we also establish a necessary and sufficient condition for the order of <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1177_Article_IEq6.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="88" /> </InlineMediaObject> <EquationSource Format="TEX">$K_{2(p-2)}(\mathcal {O}_F)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>K</mi> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mi>p</mi> <mo>-</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="script">O</mi> <mi>F</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> to be divisible by p, which generalizes a previous result of Browkin.

On the p-divisibility of even K-groups of the ring of integers of a cyclotomic field

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发表日期：

2025年8月4日

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Meng Fai Lim

摘要

Let k be a given positive odd integer and p an odd prime. In this paper, we shall give a sufficient condition for when a prime p divides the order of the groups $K_{2k}(\mathbb {Z}[\zeta _m+\zeta _m^{-1}])$ $K_{2 k} (Z [ζ_{m} + ζ_{m}^{- 1}])$ and $K_{2k}(\mathbb {Z}[\zeta _m])$ $K_{2 k} (Z [ζ_{m}])$ , where $\zeta _m$ $ζ_{m}$ is a primitive mth root of unity. When F is a p-extension contained in $\mathbb {Q}(\zeta _\ell )$ $Q (ζ_{ℓ})$ for some prime $\ell $ $ℓ$ , we also establish a necessary and sufficient condition for the order of $K_{2(p-2)}(\mathcal {O}_F)$ $K_{2 (p - 2)} (O_{F})$ to be divisible by p, which generalizes a previous result of Browkin.