**ELEMENTARY ABELIAN P-EXTENSIONS OF ALGEBRAIC FUNCTION FIELDS AND THE HASSE-ARF THEOREM**

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**Sezel Alkan**

Mathematics, MSc. Thesis, 2017

**Thesis Jury**

Assoc. Prof. Dr. Cem Güneri (Thesis Advisor), Prof. Dr. Alev Topuzoğlu,

Asst. Prof. Dr. Seher Tutdere

**Date & Time:** 06th January, 2017 – 14:00 PM

**Place: FENS 2008**

**Keywords : **Function field extension, elementary abelian extension, ramification, rational place, genus.

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**Abstract**

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This thesis starts with the basic properties of elementary abelian p-extensions of function fields. Ramification structure and the genus computation for such extensions are presented first. When the constant field is finite, number of rational places of function fields is finite and this number is bounded by the Hasse-Weil bound. However for large genus, this bound is weak. Therefore, when a sequence of function field extensions with growing genera is considered, the growth of the ratio of the number of rational places to the genera in the sequence is of interest. Following the work of Frey-Perret-Stichtenoth, we show that the limit of this ratio is zero if a sequence of abelian p-extensions are considered. Hasse-Arf theorem gives information about the jumps in the higher ramification group filtration of a function field extension. We also present the proof of this theorem for elementary abelian p-extensions, which is due to Garcia and Stichtenoth.