Timeline for Looking for a reference: double orthogonal complement in $(\mathbb{Z}/q\mathbb{Z})^n$
Current License: CC BY-SA 3.0
11 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Dec 7, 2013 at 21:28 | vote | accept | Michael Blondin | ||
| Nov 29, 2013 at 14:28 | comment | added | David Wilding | It doesn't look like Washington shows that $\widehat{A\times B}\cong\widehat A\times\widehat B$, but the fact is certainly mentioned in the proof of Lemma 3.1 ($G\cong\widehat G$ noncanonically). Many thanks. | |
| Nov 29, 2013 at 14:12 | comment | added | KConrad | OK. And earlier in that chapter Washington should have shown that $\widehat{A \times B} \cong \widehat{A} \times \widehat{B}$. Coupling the way this isomorphism works with the standard isomorphism of ${\mathbf Z}/q{\mathbf Z}$ with its character group by $a \bmod q \mapsto [b \bmod q \mapsto e^{2\pi iab/q}]$ shows $({\mathbf Z}/q{\mathbf Z})^n$ is isomorphic to its character group by the mapping I wrote down in my comment to Michael's question. | |
| Nov 29, 2013 at 14:02 | comment | added | David Wilding | Proposition 3.4 itself says that $(H^\perp)^\perp=H$, and the preceding setup is that $H$ is a subgroup of a finite abelian group $G$. $H^\perp$ is defined to be $\{\chi\in\widehat G:\chi(h)=1,\forall h\in H\}$, so I think this matches up with your comment on the question. | |
| Nov 29, 2013 at 13:54 | comment | added | KConrad | I don't have Washington's book on me at the moment and can't check it in Google books, so I can't confirm (or deny) that's right just from the proposition number. Can you write down the content of that proposition? | |
| Nov 29, 2013 at 13:49 | history | edited | David Wilding | CC BY-SA 3.0 |
Removed the claim that the result is not written anywhere.
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| Nov 29, 2013 at 13:47 | comment | added | David Wilding | Thanks @KConrad, that's very helpful. If I understand correctly, I believe you are referring to Proposition 3.4 in Introduction to Cyclotomic Fields. | |
| Nov 29, 2013 at 12:37 | comment | added | KConrad | It's definitely written down somewhere. This is a special case of a fact about characters of finite abelian groups. See, for instance, chapter 2 of Larry Washington's Introduction to Cyclotomic Fields. | |
| Nov 29, 2013 at 12:22 | review | Late answers | |||
| Nov 29, 2013 at 12:31 | |||||
| Nov 29, 2013 at 12:07 | history | answered | David Wilding | CC BY-SA 3.0 |