Announcements


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.

Forum

General


Description

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

Semester

Spring 2008 (Monday, January 21 to Friday, March 14; Monday, April 28 to Friday, May 16)

Timetable

  • Thu 12:05 - 12:55

Prerequisites

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

Syllabus

- 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

Lecturer


Ivan Fesenko
Email ivan.fesenko@nottingham.ac.uk
Phone (0115) 9514952
Interests Number theory
Photo of Ivan Fesenko


Students


Photo of Wemedh Aeal
Wemedh Aeal
(Manchester)
Photo of Oliver Braeunling
Oliver Braeunling
(Nottingham)
Photo of Ben Fairbairn
Ben Fairbairn
(Birmingham)
Photo of Almar Kaid
Almar Kaid
(Sheffield)
Photo of Jonathan Mason
Jonathan Mason
(Nottingham)
Photo of Richard Slessor
Richard Slessor
(Southampton)
Photo of Paul Truman
Paul Truman
(Exeter)
Photo of Panagiotis Tsaknias
Panagiotis Tsaknias
(Sheffield)


Bibliography


No bibliography has been specified for this course.

Assessment



No assessment information is available yet.

No assignments have been set for this course.

Files


Files marked L are intended to be displayed on the main screen during lectures.

Week(s)File
he.pdfL
l1.pdfL
l10.pdfL
l2.pdfL
l3.pdfL
l4.pdfL
l5.pdfL
l6.pdfL
l7.pdfL
l8.pdfL
l9.pdfL
lf.pdfL


Recorded Lectures


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