## Abstract

NTP_{2} is a large class of first-order theories defined by Shelah generalizing simple and NIP theories. Algebraic examples of NTP_{2} structures are given by ultra-products of p-adics and certain valued difference fields (such as a non-standard Frobenius automorphism living on an algebraically closed valued field of characteristic 0). In this note we present some results on groups and fields definable in NTP_{2} structures. Most importantly, we isolate a chain condition for definable normal subgroups and use it to show that any NTP_{2} field has only finitely many Artin-Schreier extensions. We also discuss a stronger chain condition coming from imposing bounds on burden of the theory (an appropriate analogue of weight) and show that every strongly dependent valued field is Kaplansky.

Original language | English |
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Pages (from-to) | 395-406 |

Number of pages | 12 |

Journal | Proceedings of the American Mathematical Society |

Volume | 143 |

Issue number | 1 |

DOIs | |

State | Published - 1 Jan 2015 |

### Bibliographical note

Publisher Copyright:© 2014 American Mathematical Society.

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