We consider Shanks’ simplest cubic fields K for which the index <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1005_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="77" /> </InlineMediaObject> <EquationSource Format="TEX">$[{\mathcal {O}}_K:{\mathbb {Z}}[\rho ]]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <msub> <mi mathvariant="script">O</mi> <mi>K</mi> </msub> <mo>:</mo> <mi mathvariant="double-struck">Z</mi> <mrow> <mo stretchy="false">[</mo> <mi>ρ</mi> <mo stretchy="false">]</mo> </mrow> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> of a root <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1005_Article_IEq2.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\rho $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ρ</mi> </math></EquationSource> </InlineEquation> of the defining parametric polynomial is 3. For them, we study the additive indecomposables of K and provide a complete list of them. Moreover, we use the knowledge of the indecomposables to prove some interesting consequences on the arithmetic of K. Mainly, we obtain good bounds on the ranks of universal quadratic forms over K and prove that the Pythagoras number of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1005_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="27" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {O}}_K$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">O</mi> <mi>K</mi> </msub> </math></EquationSource> </InlineEquation> is 6.

Additive structure of non-monogenic simplest cubic fields

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2025年1月31日

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Daniel Gil-Muñoz,
Magdaléna Tinková

摘要

We consider Shanks’ simplest cubic fields K for which the index $[{\mathcal {O}}_K:{\mathbb {Z}}[\rho ]]$ $[O_{K} : Z [ρ]]$ of a root $\rho $ $ρ$ of the defining parametric polynomial is 3. For them, we study the additive indecomposables of K and provide a complete list of them. Moreover, we use the knowledge of the indecomposables to prove some interesting consequences on the arithmetic of K. Mainly, we obtain good bounds on the ranks of universal quadratic forms over K and prove that the Pythagoras number of ${\mathcal {O}}_K$ $O_{K}$ is 6.