1.  Introduction:   Taught by lectures, together with extensive individual and small group consultation. This unit is in the Master of Maths degree programme and also contains material suitable for postgraduate students.

2.  Timetabled Hours, Credits, Assessment:    This is a 20 UCU unit, given in the Spring Semester with 22 lectures, and time set aside for student-lecturer consultations. The assessment is by 20% coursework and 80% examination. The coursework consists of several sets of exercises which are marked and returned with feedback and model solutions. The exam paper is 3 hours consisting of 4 questions.

3.  Overview:   The earliest examples of local fields were the p-adic numbers Q_p, with p prime, introduced by Hensel (1902), by analogy with power series. They are closely related to the problem of solving congruences modulo p. For example, the congruence x²=2 (mod 7ⁿ) has an integer solution for any value of n but there is no simultaneous solution; but if we allow power series in p (in the example p=7), like

a₀+a₁p+a₂p²+ a₃p³+...

then we can get such a solution. Of course, with the usual absolute value, these power series don't converge, so we must introduce a different absolute value (the p-adic absolute value) to make sense of this.

A general notion of absolute value was introduced by Kurschak (1913) and the set of all valuations of the rationals was determined by Ostrowski (1918), which we will see.

One big problem with doing analysis in the rationals is that Q is not complete - there are sequences which ought to converge but which don't. When we are considering the usual absolute value, we get round this problem by passing to the reals R. In the same way, we can construct the completion of Q with respect to the p-adic absolute value, and this is how we get Q_p. Analysis here turns out to be even easier than on the reals.

Now the problem with the field of real numbers R is that it is not algebraically closed - there are polynomials with coefficients in R which do not have roots in R. We get round this by extending the field to get the complex numbers C. In fact, the situation here is remarkably simple: we only have to worry about getting the root to one particular polynomial (namely x²+1) and then it turns out we have them all! (Another way of saying this is that C/R is an extension of degree 2.)

For the p-adic numbers, matters are a little more complicated. We will still define the algebraic closure of Q_p, but it is an extension of infinite degree. To understand it better we can look at the finite subextensions, which turn out to be fields with remarkably similar properties to Q_p. Trying to understand the way in which Q_p is extended leads to a large and fascinating theory.

Finally, the field of complex numbers C really is the right place to carry out analysis because it is still complete. Unfortunately, the algebraic closure of Q_p is not complete so we have to complete it again, obtaining C_p. Miraculously, this is still algebraically closed, so C_p is the place to do p-adic analysis - though p-adic analysis turns out to have a rather different flavour to real or complex analysis!

It would be useful to remind yourselves of some of the language of rings and fields from Algebra II. Books for this include:

Books for the course itself include:

We will aim to cover some or all of the following topics: