MAGIC037: Local fields

Course details

A core MAGIC course


Spring 2024
Monday, January 29th to Friday, March 22nd; Monday, April 22nd to Friday, May 3rd


Live lecture hours
Recorded lecture hours
Total advised study hours


10:05 - 10:55
13:05 - 13:55

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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
For a much more comprehensive source, a book on local fields (S.V. Vostokov, I.B. Fesenko) see


some basic knowledge of p-adic numbers will be useful; read, e.g.,
4 pages of part 4 of - Introduction to number theory, 5th semester course
and pp.37-41 of part 4 of - 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


  • CW

    Chris Wuthrich

    University of Nottingham
    Main contact
  • CW

    Chris Williams

    University of Nottingham


No bibliography has been specified for this course.


The assessment for this course will be released on Monday 13th May 2024 at 00:00 and is due in before Friday 24th May 2024 at 11:00.

Assessment for all MAGIC courses is via take-home exam which will be made available at the release date (the start of the exam period).

You will need to upload a PDF file with your own attempted solutions by the due date (the end of the exam period).

If you have kept up-to-date with the course, the expectation is it should take at most 3 hours’ work to attain the pass mark, which is 50%.

Please note that you are not registered for assessment on this course.


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