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Elementarna ekvivalenca polj z valuacijami in izrek Ax-Kochen-Jeršov : magistrsko delo
Vukšić, Lara (Author), Klep, Igor (Mentor) More about this mentor... This link opens in a new window

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Abstract
Valuacija je homomorfizem, ki slika multiplikativno grupo obrnljivih elementov polja v urejeno abelovo grupo. Če valuacija slika v aditivno grupo celih števil, je diskretna. Chevalleyev izrek nam pove, da lahko vsako valuacijo polja razširimo tudi na nadpolja. Če za vsako algebraično razširitev polja z valuacijo obstaja natanko ena razširitev valuacije, pravimo, da je polje Henselovo. Primeri Henselovih polj so polna polja z diskretno valuacijo. Dve strukturi v jeziku sta elementarno ekvivalentni natanko tedaj, ko vsak stavek v tem jeziku velja v eni natanko tedaj, ko velja v drugi. Vse izomorfne strukture so elementarno ekvivalentne, obratno pa v splošnem ne velja. Izrek Ax-Kochen-Jeršov za pare Henselovih polj z valuacijo natančno pove, kdaj so elementarno ekvivalentni. Po njegovi posledici vsak stavek velja v polju $p$-adičnih števil ${\mathbb Q}_p$ za skoraj vsa praštevila p natanko tedaj, ko velja v polju Laurentovih vrst ${\mathbb Z}_p((t))$ za skoraj vsa praštevila $p$.

Language:Slovenian
Keywords:polja z valuacijo, urejene abelove grupe, napolnitev polja z valuacijo, razširitve valuacij, Henselova polja z valuacijo, elementarna ekvivalenca, elementarne razširitve, tipi, nasičene strukture, izrek Ax-Kochen-Jeršov
Work type:Master's thesis/paper (mb22)
Tipology:2.09 - Master's Thesis
Organization:FMF - Faculty of Mathematics and Physics
Year:2018
UDC:510.6
COBISS.SI-ID:18431577 Link is opened in a new window
Views:266
Downloads:135
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Secondary language

Language:English
Title:Elementary equivalence of valued fields and Ax-Kochen-Eršov theorem
Abstract:
A valuation on a field is a homomorphic mapping from the multiplicative group of invertible elements of a field into an ordered abelian group. If it maps into the additive group of integers, it is called discrete. By Chevalley’s theorem, every valuation on a field extends to any field extension. Henselian valued fields are those for which valuation extends uniquely to any algebraic field extension. For example, complete discrete valued fields are Henselian. Two structures of a given language are elementary equivalent if and only if every sentence in this language holds in the first structure if and only if it also holds in the second. All isomorphic structures are elementary equivalent, but the converse is not true in general. The Ax-Kochen-Eršov theorem explains when any two Henselian valued fields are elementary equivalent. As a consequence, a sentence holds in the field of $p$-adic numbers ${\mathbb Q}_p$ for almost all primes $p$ if and only if it holds in the field of Laurent series ${\mathbb Z}_p((t))$ for almost all primes $p$.

Keywords:valued fields, ordered abelian groups, completion of valued field, extensions of valuations, Henselian valued fields, elementary equivalence, elementary extensions, types, saturated structures, Ax-Kochen-Eršov theorem

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