### Algebraic Function Fields (MATH 636)

2022 Fall
Faculty of Engineering and Natural Sciences
Mathematics(MATH)
3
10
Nurdagül Anbar Meidl nanbar@sabanciuniv.edu,
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English
Doctoral, Master
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### CONTENT

Places,valuation rings and discrete valuations of a function field; the rational function field; divisors, Weil different adeles, genus; Riemann-Roch Theorem and its consequences; extensions of function fields, ramification, Hurwitz genus formula; constant field extensions, Galois extensions, Kummer and Artin-Schreier extensions.

### LEARNING OUTCOMES

• - discuss the basic notions of the theory, like places, divisors, Riemann-Roch spaces, and genus of a function field.
• - recall the main results, centered around the Riemann-Roch theorem.
• - recall the notions of ramification and splitting in extensions of function fields.
• - apply these concepts in specific cases, in particular fields of small genus.

### ASSESSMENT METHODS and CRITERIA

 Percentage (%) Final 30 Midterm 30 Individual Project 40

### RECOMENDED or REQUIRED READINGS

 Textbook Algebraic Function Fields and Codes, by Henning Stichtenoth, 2nd Edition, Springer-Verlag, 2009. Graduate Texts in Mathematics No. 254.