Timeline for Is there a notion of integration over the algebraic numbers?
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| Apr 19, 2010 at 14:47 | comment | added | Keenan Kidwell | @Pete Yeah, I thought about that as soon as I posted my response. I'm also not really sure what Jose has in mind, but it's good to point this out, that there is one obvious locally compact topology on $\overline{\mathbb{Q}}$. | |
| Apr 19, 2010 at 14:32 | comment | added | Pete L. Clark | @KK: As with the integers, you could put the discrete norm on $\overline{\mathbb{Q}}$ and then it becomes locally compact and complete, with natural (Haar) measure being the counting measure. This feels like a rather trivial answer, but I can't tell from the question exactly what Jose has in mind. | |
| Apr 19, 2010 at 11:59 | comment | added | Keenan Kidwell | The fact that the set of algebraic numbers is countable is not what precludes applying the usual integration machinery to it. The additive group of integers is a perfectly good (countable) locally compact group and people do Fourier analysis on it all the time. The problem with $\overline{\mathbb{Q}}$ is that it fails to be locally compact in any of its usual topologies. | |
| Apr 19, 2010 at 11:57 | history | edited | Gerald Edgar | CC BY-SA 2.5 |
added 4 characters in body; edited body
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| Apr 19, 2010 at 11:52 | history | answered | Gerald Edgar | CC BY-SA 2.5 |