Abstract /
résumé
Name / nom: Erica Blom
School / école: Queen's University at Kingston
Length / durée: 50 min
Title / titre: Not-so-sporadic p-adics: Completions of Q
Abstract / résumé:
What every math student knows is that the real numbers, R, complete the rationals,
Q. What slightly fewer math students know is that R is but one of many completions of
Q; in fact, there are infinitely many completions of Q. These include the p-adic
numbers (where p denotes some prime); if we let p be infinity, we obtain
the real numbers. Ostrowski's Theorem claims that in fact the p-adic numbers
are, up to equivalence, the *only* completions of Q.
But what *are* p-adic numbers? This talk aims to provide a sufficient
introduction to these nefarious objects such that Ostrowski's Theorem may be
appreciated. I will discuss various things concerning the p-adic numbers,
including the ring of p-adic integers, p-adic valuation, the
p-adic ultrametric, the isosceles triangle property, and a few applications
(time permitting).
Prerequisites / choses nécessaires: Some analysis (i.e. norms and
metrics) and some algebra (i.e. basic notion of a ring and a fraction field).
PDF format / format PDF: blom.pdf