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<p>We investigate the model completeness of the theory of a mixed characteristic henselian valued field with finite ramification relative to the model completeness of the residue field and value group. We address the case in which the valued field has a value group with finite spines, and the case in which the value group is elementarily equivalent to the infinite lexicographic sum of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {Z}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">Z</mi> </math></EquationSource> </InlineEquation> with a minimal positive element. In both cases, we find a one-sorted language in which the theory of the valued field is model complete, if the theory of the residue field is model complete in the language of rings.</p>

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Relative model completeness of henselian valued fields with finite ramification and various value groups

  • 期刊论文

  • 发表日期：

    2026年5月25日

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  • Anna De Mase

摘要

We investigate the model completeness of the theory of a mixed characteristic henselian valued field with finite ramification relative to the model completeness of the residue field and value group. We address the case in which the valued field has a value group with finite spines, and the case in which the value group is elementarily equivalent to the infinite lexicographic sum of \(\mathbb {Z}\) Z with a minimal positive element. In both cases, we find a one-sorted language in which the theory of the valued field is model complete, if the theory of the residue field is model complete in the language of rings.