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A talk on the p-adic numbers
*September 16, 2009*

*Posted by Akhil Mathew in algebraic number theory, General, math education, number theory.*

Tags: independent study, p-adic numbers, talks

7 comments

Tags: independent study, p-adic numbers, talks

7 comments

The start of the academic year has made it much more difficult for me to get in serious posts as of late, and the number theory series has slowed. Things should clear up at least somewhat in a few more weeks. In the meantime, I’ll do something that occurred to me a while back but I then forgot about: posting a talk.

I took an independent study course last semester on class field theory. As is traditional, I gave a talk last May after the course on some aspects of the subject matter. Several faculty members at the university and teachers in my school attended, along with some undergraduates there. In the talk, I gave an elementary overview of the p-adic numbers, assuming no more than basic number theory and point-set topology.

Anyway, I am posting the (slightly corrected) presentation and the notes here.

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Completions of fields
*September 1, 2009*

*Posted by Akhil Mathew in algebra, algebraic number theory, commutative algebra, number theory.*

Tags: absolute values, Cauchy sequences, completions, p-adic numbers

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Tags: absolute values, Cauchy sequences, completions, p-adic numbers

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So again, we’re back to completions, though we’re going to go through it quickly. Except this time we have a field with an absolute value like the rationals with the usual absolute value.

**Completions **

Definition 1The(more…)completionof is defined as the set of equivalence classes of Cauchy sequences: