p-adic numbers, January-February 2010
For PhD students at the London Taught Course Centre.
Course description: The aim of this course is to explain the construction and basic properties of
p-adic numbers and to give an introduction to some elementary aspects of p-adic analysis.
Syllabus: absolute values, completions, the fields Qp and Cp,
normed spaces, continuous functions, Mahler's expansion, differentiable functions, p-adic versions
of some classical functions (ax, Γp, ...)
Lectures:
Time: Mondays 10.30-12.30, 18 January 2010 to 15 February 2010
Location: De Morgan House, Russell Square
Lecturer: Dr Manuel Breuning,
King's College London
Course materials:
Lecture notes, Chapter 1
(Absolute values and completion)
Lecture notes, Chapter 2
(The fields Qp and Cp)
Lecture notes, Chapter 3
(Normed spaces)
Lecture notes, Chapter 4
(Continuous functions on Zp)
Lecture notes, Chapter 5
(Differentiation)
Lecture notes
(Chapters 1-5 in one file; includes some minor corrections)
Solutions to selected exercises
(Exercises 1.13, 1.16, 2.10, 2.23) If you need solutions/hints for any other exercises, please email me.
Mock exam
Solutions to the mock exam
Exam
Solutions to the exam
Books:
- J.W.S. Cassels: Local fields. Cambridge University Press, 1986.
- F.Q. Gouvea: p-adic numbers. 2nd edition, Springer, 1997.
- N. Koblitz: p-adic numbers, p-adic analysis, and zeta-functions. 2nd edition,
Springer, 1984.
- A.M. Robert: A course in p-adic analysis. Springer, 2000.
- W.H. Schikhof: Ultrametric calculus. Cambridge University Press, 1984.
Previous LTCC courses:
Algebraic Number Theory (January-February 2008)
Algebraic Number Theory (January-February 2009)
last modified 3 June 2010