Download p-adic Numbers

Supervisors Project Submission Form
Supervisor:
M D Coleman
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Project Title
p-adic Numbers
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Category
Pure
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Level
3/4
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Semesters
1/2
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Description
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References
The set of p-adic integers Zp can be introduced in two ways.
[a] The first would be as in infinite sequence of p-adic digits,
indexed by N, with rules to add and multiply such sequences.
This would lead to a ring structure, from which the field of
fractions can be taken as the definition of p-adic numbers, Qp.
This should be reminiscent of Q being the field of fractions of
Z. Having concrete realizations of p adic integers means that
every step can be better understood and illustrated by nontrivial worked examples and this can enhance any project.
[b] Alternatively, each infinite sequence of p-adic digits can be
taken as the coefficients in an infinite series in powers of p.
The question of how such series converge then arises. Such
series cannot converge with respect to the usual distance
function on integers. Instead we have to consider what are
called p-adic valuations. These valuations are first defined on
the integers and then the rationals. The set of integers is then
completed with respect to the p-adic valuation to give Z_p
while the rationals are completed to give Qp.
Such an approach requires good understanding of analysis.
Part of a project would be to show that the Zp and Qp given
by both approaches [a] and [b] are the same.
A possible goal of a project would be Hensel's Lemma (a basic
form is found in [4] Section 2.6). This result has many
applications. For instance, in Chapter 4 of [3] the existence of
local and global solutions of Diophantine equations is studied.
1. Borevich, Shafarevich, Number Theory, [Chapter 1]
2. Bachmann, Introduction to p-adic numbers and valuation
theory.
3. Cassels, J.W.S, Local Fields LMS Student Texts 3.
4. Niven, Zuckerman, Montgomery, An Introduction to the
Theory of Numbers, 5th edition, Wiley, 1991.
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Prerequisite courses
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Additional notes