Local fields (MAGIC037)
Examination for this course is in the form of written essay on several topics. Specific topics will be supplied individually to students willing to take an exam.
On one hand, local fields form the class of fields which is the next most easiest to study after the class of finite fields, and hence they are quite useful for and applicable in many parts of mathematics, on the other hand, local fields show up in the local study of various parts of mathematics including number theory, algebraic geometry, algebraic topology and areas of mathematical physics. This general and very short course will discuss the main examples, features and type of behaviour of local fields and local arithmetic. The lecture notes of the course are available from http://www.maths.dept.shef.ac.uk/magic/course_files/37/lf.pdf For a much more comprehensive source, a book on local fields (S.V. Vostokov, I.B. Fesenko) see http://www.maths.nott.ac.uk/personal/ibf/book/book.html
Spring 2008 (Monday, January 21 to Friday, March 14; Monday, April 28 to Friday, May 16)
some basic knowledge of p-adic numbers will be useful; read, e.g., 4 pages of part 4 of http://www.maths.nott.ac.uk/personal/ibf/num/num.pdf - Introduction to number theory, 5th semester course and pp.37-41 of part 4 of http://www.maths.nott.ac.uk/personal/ibf/aln/aln.pdf - Introduction to algebraic number theory, 6th semester course
- discrete valuations, discrete valuation fields, completion - norms on Q - local fields - additive and multiplicative topological structures of a local field - Henselian property - nonramified extensions of local fields - tamely ramified extensions of local fields - wildly ramified extensions of local fields, ramification groups filtration - invariants associated to the norm map for finite extensions of local fields - explicit reciprocity map - main theorems of the local class field theory - the Hilbert symbol and explicit formulae
No bibliography has been specified for this course.
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