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| ``Re: Imaginary irrational number?''
by joking on 2007-04-16 04:38:04 |
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| > I certainly concede that every field contains a subring naturally
> isomorphic to the integers, but for example, in the characteristic 0
> field of the p-adic rationals, this definition does not coincide with
> the p-adic integers. In C_p, the complex p-adics, a realm in which we
> would actually like to make a distinction between rational and
> irrational, the distinction is even more exacerbated.
Every field has a simple field as subfield, i.e. smallest (inclusion) subfield of F. If a characteristic of a field is p>0, then this subfield is isomorphic to Z_{p} (notice that p is prime). If a characteristic is 0, then this subfield is isomorphic to Q.
So if you consider any field F of characteristic 0, then you can assume that Q is a subfield of F. There's even more - there's no other subfield of F isomorphic to Q (since every subfield of F contains Q and Q is simple). So then definiton makes sense:
Irrational element (yeah... this is my definition :) ) in char 0 field F is any element in F\Q.
Irrational number is irrational element in R = real numbers.
Also C_{p} is isomorphic (as fields) with C = complex numbers. Of course Q_{p} can be embedded in C_{p}, so Q_{p} is nothing else then subfield of C = complex numbers. So I don't understand where's the problem?
Of course all of this is as a bit strange. I don't know whether there's use of irrational elements (instead of numbers).
joking |
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