Timeline for checking if F[x]/I is isomorphic to F[x]/J
Current License: CC BY-SA 3.0
16 events
| when toggle format | what | by | license | comment | |
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| May 17, 2011 at 4:56 | comment | added | Amritanshu Prasad | Thanks Felipe. Indeed, it appears in Chapter II of Serre's Corps Locaux. | |
| May 17, 2011 at 2:43 | vote | accept | CommunityBot | ||
| May 16, 2011 at 11:13 | comment | added | Felipe Voloch | The fact that (for $p(x) \in F[x]$ irreducible) $F[x]/p^n$ is isomorphic to $E[t]/t^n, E = F[x]/p$ is (essentially equivalent to and) a consequence of the fact that the completion of $F(x)$ at the place $p$ is isomorphic to the power series field $E((t))$. This is the well-known characterization of equicharacteristic complete discretely valued fields. This you can find in most books on local fields, e.g., Serre's. | |
| May 16, 2011 at 7:13 | history | edited | Amritanshu Prasad | CC BY-SA 3.0 |
Changed link, since html rendering was poor, with $f'$ appearing as $f$ often.
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| May 16, 2011 at 7:06 | comment | added | Amritanshu Prasad | Good point: you decompose $F[x]/I$ into a sum of indecomposable ideals (this coincides with the primary decomposition and is unique). Also, maybe I should not have said $p$ when I meant $p_i$. | |
| May 16, 2011 at 7:04 | history | edited | Amritanshu Prasad | CC BY-SA 3.0 |
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| May 16, 2011 at 6:49 | comment | added | user5810 | re edit: We still need to show that the factors ($E_i$ and $n_i$) can be recovered from $F[x]/I$. | |
| May 16, 2011 at 6:17 | history | edited | Amritanshu Prasad | CC BY-SA 3.0 |
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| May 16, 2011 at 6:02 | history | edited | Amritanshu Prasad | CC BY-SA 3.0 |
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| May 16, 2011 at 6:01 | comment | added | Amritanshu Prasad | "only if" does need some more work. | |
| May 16, 2011 at 5:52 | comment | added | Qiaochu Yuan | Oh, you also need to transport $F$, I guess. | |
| May 16, 2011 at 5:52 | comment | added | Qiaochu Yuan | @Ricky: for finite fields I think you can do it by counting the number of roots of various polynomials and induction. Alternately, you can apply the structure theorem to modules over $F[x]$ (since if $\phi : F[x]/I \to F[x]/J$ is an isomorphism of rings you get an isomorphism of $F[x]$-modules by letting $x$ act as $\phi(x)$ on $F[x]/J$.) | |
| May 16, 2011 at 5:48 | comment | added | user5810 | How do you get "and only if"? | |
| May 16, 2011 at 5:30 | history | edited | Amritanshu Prasad | CC BY-SA 3.0 |
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| May 16, 2011 at 5:22 | history | edited | Amritanshu Prasad | CC BY-SA 3.0 |
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| May 16, 2011 at 4:58 | history | answered | Amritanshu Prasad | CC BY-SA 3.0 |