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**Dp-minimal valued fields.**
*(English)*
Zbl 1385.03040

In this paper, the authors investigate the algebraic structure of dp-minimal valued fields. Dp-minimal theories can be thought of as a one-dimensional version of NIP theories; specifically, they are the theories in which the dp-rank of every type in one variable is at most one. Dp-minimality is a common generalization of C-minimality, o-minimality, and P-minimality, and hence the class of dp-minimal valued fields includes algebraically closed, real closed, and \(p\)-adically closed fields.

In [Fun with fields. Berkeley, CA: University of California (Diss.) (2016)], W. A. Johnson classified all dp-minimal fields up to elementary equivalence (see also [“On dp-minimal fields”, Preprint, arXiv:1507.02745]). Johnson constructs a valuation on any dp-minimal field that is not strongly minimal, and shows that the constructed valuation is always Henselian. This paper was developed independently of Johnson’s work [loc. cit.], and proves many of the same results using different methods. The authors provide a detailed comparison between their work and Johnson’s work [loc. cit.] in the introduction.

Section 3 of this paper generalizes results of J. Goodrick [J. Symb. Log. 75, No. 1, 221–238 (2010; Zbl 1184.03035)] and P. Simon [J. Symb. Log. 76, No. 2, 448–460 (2011; Zbl 1220.03037)] from the setting of dp-minimal ordered structures to the setting of dp-minimal structures with a definable uniform structure. A uniform structure is a collection of subsets which are able to replace intervals in the results of Goodrick [loc. cit.] and Simon [loc. cit.]; when considering valued fields, the uniform structure will be a collection of balls, that is, sets of the form \(\{x\in K : v(x-a) > \gamma\}\).

The main results of Section 3 are Propositions 3.6 and 3.9: if \(M\) is a structure with both a divisble abelian group operation and a uniform structure (for instance, if \(M\) is a valued field), then every infinite definable subset of \(M\) has non-empty interior and the image of any set with non-empty interior under a definable finite-to-one function also has non-empty interior.

Section 4 uses the results of Section 3 to prove the main result of the paper: every dp-minimal valued field is Henselian. Finally, Sections 5 and 6 classify dp-minimal ordered fields up to elementary equivalence: any such field is elementarily equivalent to a Hahn series field with residue field \(\mathbb R\) and nonsingular value group (meaning \(\Gamma/p\Gamma\) is finite for all primes \(p\)).

The paper is very accessible, and only assumes a passing familiarity with dp-minimality. The content on valuation theory makes more demands of the reader, but still clearly states and references any results that are not proved within the paper.

In [Fun with fields. Berkeley, CA: University of California (Diss.) (2016)], W. A. Johnson classified all dp-minimal fields up to elementary equivalence (see also [“On dp-minimal fields”, Preprint, arXiv:1507.02745]). Johnson constructs a valuation on any dp-minimal field that is not strongly minimal, and shows that the constructed valuation is always Henselian. This paper was developed independently of Johnson’s work [loc. cit.], and proves many of the same results using different methods. The authors provide a detailed comparison between their work and Johnson’s work [loc. cit.] in the introduction.

Section 3 of this paper generalizes results of J. Goodrick [J. Symb. Log. 75, No. 1, 221–238 (2010; Zbl 1184.03035)] and P. Simon [J. Symb. Log. 76, No. 2, 448–460 (2011; Zbl 1220.03037)] from the setting of dp-minimal ordered structures to the setting of dp-minimal structures with a definable uniform structure. A uniform structure is a collection of subsets which are able to replace intervals in the results of Goodrick [loc. cit.] and Simon [loc. cit.]; when considering valued fields, the uniform structure will be a collection of balls, that is, sets of the form \(\{x\in K : v(x-a) > \gamma\}\).

The main results of Section 3 are Propositions 3.6 and 3.9: if \(M\) is a structure with both a divisble abelian group operation and a uniform structure (for instance, if \(M\) is a valued field), then every infinite definable subset of \(M\) has non-empty interior and the image of any set with non-empty interior under a definable finite-to-one function also has non-empty interior.

Section 4 uses the results of Section 3 to prove the main result of the paper: every dp-minimal valued field is Henselian. Finally, Sections 5 and 6 classify dp-minimal ordered fields up to elementary equivalence: any such field is elementarily equivalent to a Hahn series field with residue field \(\mathbb R\) and nonsingular value group (meaning \(\Gamma/p\Gamma\) is finite for all primes \(p\)).

The paper is very accessible, and only assumes a passing familiarity with dp-minimality. The content on valuation theory makes more demands of the reader, but still clearly states and references any results that are not proved within the paper.

Reviewer: Peter Sinclair (Hamilton)

### MSC:

03C60 | Model-theoretic algebra |

03C45 | Classification theory, stability, and related concepts in model theory |

03C64 | Model theory of ordered structures; o-minimality |

12J12 | Formally \(p\)-adic fields |

12J10 | Valued fields |

12J15 | Ordered fields |

12L12 | Model theory of fields |

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\textit{F. Jahnke} et al., J. Symb. Log. 82, No. 1, 151--165 (2017; Zbl 1385.03040)

### References:

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