Timeline for Generators of prime ideals and factorization of polynomials
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Oct 3, 2019 at 13:07 | comment | added | Aurel | Yes, and if you want examples with the property you mention for all $p$, you can take $X^4-8$ for $p=2$ and $X^p-p^2$ for $p$ odd. | |
| Oct 3, 2019 at 13:03 | history | edited | Aurel | CC BY-SA 4.0 |
minor extra hypothesis
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| Oct 3, 2019 at 11:09 | comment | added | Algebrus | ... the factorization patterns of $f$ (mod $3$) and $3O_K$ agree, but the unique prime lying over $3$ in $O_K$ (namely, the principal ideal ($3^{1/3}$) is not of the desired shape. | |
| Oct 3, 2019 at 11:07 | comment | added | Algebrus | Thank you for providing intuition and example. Of particular interest is the case p=3, because then both $\mathrm{disc}(K)$ (for $K = \mathbb{Q}[X]/(f)$) and $\mathrm{disc}(f)$ are (distinct) powers of $3$, thus also the index $[O_K : \mathbb{Z}[X]/(f)]$ is a (non-trivial) power of $3$, as indicated by @KConrad here: mathoverflow.net/questions/21247/…. (In particular, $O_K$ is not generated by $f$ over $\mathbb{Z}$.) Thus 3 is the only prime not covered by the Kummer-Dedekind-Theorem, and as you indicate, ... | |
| Oct 3, 2019 at 11:01 | vote | accept | Algebrus | ||
| Oct 3, 2019 at 8:38 | history | edited | Aurel | CC BY-SA 4.0 |
more precise formulation
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| Oct 3, 2019 at 8:24 | history | answered | Aurel | CC BY-SA 4.0 |