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| ``Re: Imaginary irrational number?''
by alozano on 2007-04-16 10:03:07 |
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| mathcam said: --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.--
Definitely. The issue here is that the term rational has several meanings. The term rational (as in rationals and integers) has a different meaning than rational (as in rational and irrational). The "integers" of the "rationals" Q_p are the elements of Z_p, but Z_p, for certain p, contains elements of C (complex numbers) which are definitely not rational (as in "irrational"). For example, Z_5 contains i=sqrt(-1), or Z_7 contains sqrt(2). My point then is that even though we do want to call Z_p the "integers" of Q_p, the elements of Z_p are not necessarily what we want to call "rational" (not irrational) inside C_p.
Alvaro |
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