Algebraic function fields

at the University of Antwerp • winter term 2024 • by Ingo Blechschmidt

🎨 Exercises

• Sheets
• Solutions
• Agda templates • transcripts of our Agda sessions • Agdapad • more resources on Agda

💡 Syllabus

-1. The function field analogy
0. Algebraic preliminaries
   rings, integral domains, local rings, fields • polynomial rings • localization • ideals, quotients by ideals, charts of ideals, principal ideal domains • unique factorization domains • algebraic closure
1. Algebraic sets
   affine space • zero loci and nonzero loci with examples and basic properties • defining ideals and their reconstruction
2. Affine varieties
   irreducible algebraic sets • varieties in two-dimensional affine space • Nullstellensatz corollaries • Noether normalization and dimension • rings of germs • coordinate-free algebraic geometry • normal varieties
3. Function fields in one variable
   definition • generators

Rough transcript.

📖 References

• William Fulton. Algebraic curves (pdf), 2008.
• Bjorn Poonen. Lectures on rational points on curves (pdf), 2006.
• Renate Scheidler. Algebraic number theory in function fields (pdf), 2016.
• Henning Stichtenoth. Algebraic function fields and codes, 2009.

💬 Contact

mail: iblech-antwerp@speicherleck.de
phone: +49 176 95110311 (also Telegram)