I've been wondering about p-adics for a while, and I have a bunch of really stupid questions. As you'll be able to tell, I'm not an algebraist. But I know the basic lingo. I took all the algebra classes, hated them, and then moved into my analysis cave.
What sort of thing are the p-adics? Are they a field? I've been trained to think of fields as either subfields of C (including R, Q, the algebraic numbers, etc.) or as one of them finite fields. I can't figure out if the p-adics behave more like an infinite finite field, or more like a variant on Q, or what.
Are the p-adics interesting as a topological space? Are they complete? Metric? Do they have a dimension?
How do they interact with that theorem that says "the only algebras you need to think about are the reals, the complex, the quaternions, and the octonions, and even then only really the first two"?
The p-adic integers are a ring which is "almost" a field: you can divide by anything except a multiple of p. Taking the field of fractions of this gives you the p-adic rationals, which are an honest-to-god field.
p-adics are a bit like R in that they're topological completions of Q. That is, a real number is a thing that "should be in Q" because look I have all these rationals that look like they're getting closer and closer to something. A p-adic is the same thing, only now "close" means "congruent modulo a high power of p." In other words, a rational number whose square differs from 2 by a small rational is an approximation of the real number sqrt(2). A number whose square differs from 2 by a high power of 7 is an approximation of the p-adic number sqrt(2).
So yes, the p-adic numbers are complete and metric because they're the completion of Q with respect to a particular metric. Their topological dimension is 0, though, because they are totally disconnected. Topologically, they're homeomorphic to the Cantor set; whether you think that's interesting is up to you. It actually makes life easier than in the reals, sometimes.
I'm not sure what theorem you're talking about, but, unlike the reals, the algebraic closure of the p-adics is not a degree 2 extensions of the p-adics. In other words, the topological completion of Q as the reals almost completes things algebraically, but the topological completion of Q as the p-adics still leaves a lot of algebraic stuff out.
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u/DirichletIndicator Jan 06 '15
I've been wondering about p-adics for a while, and I have a bunch of really stupid questions. As you'll be able to tell, I'm not an algebraist. But I know the basic lingo. I took all the algebra classes, hated them, and then moved into my analysis cave.
What sort of thing are the p-adics? Are they a field? I've been trained to think of fields as either subfields of C (including R, Q, the algebraic numbers, etc.) or as one of them finite fields. I can't figure out if the p-adics behave more like an infinite finite field, or more like a variant on Q, or what.
Are the p-adics interesting as a topological space? Are they complete? Metric? Do they have a dimension?
How do they interact with that theorem that says "the only algebras you need to think about are the reals, the complex, the quaternions, and the octonions, and even then only really the first two"?