Timeline for Produce an irreducible polynomial that can't be proved irreducible by using Eisenstein [closed]
Current License: CC BY-SA 3.0
9 events
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| Nov 8, 2011 at 16:40 | comment | added | paul garrett | Newton polygon scenarios systematically give examples just-slightly-more-complicated than Eisenstein-criterion examples. E.g., $x^{n+1}+2x+4$: the slopes are $1/n$ $n$ times and a single $1$. Thus, this has at least an irreducible degree-$n$ factor. Excluding a rational root is easy (not $\pm 1,\pm 2\pm 4$), so it's irreducible. | |
| Nov 8, 2011 at 12:39 | history | closed |
Felipe Voloch user6976 Martin Brandenburg Bruce Westbury Torsten Ekedahl |
too localized | |
| Nov 8, 2011 at 12:32 | history | edited | Maurizio Monge |
edited tags
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| Nov 8, 2011 at 12:28 | answer | added | Maurizio Monge | timeline score: 1 | |
| Nov 8, 2011 at 11:59 | comment | added | KConrad | It would be nice if you didn't formulate this as a command ("give an example...") and explained why you are asking (idle curiosity, homework,...). Your question isn't at the intended level of MO, but I'll make a comment which I think is: if $K$ is a number field in which (1) the ring of integers has the form ${\mathbf Z}[\alpha]$ and (2) no prime number is totally ramified, then the minimal polynomial of $\alpha$ over ${\mathbf Q}$ has the feature you seek. Many cyclotomic extensions of ${\mathbf Q}$ fits these properties. | |
| Nov 8, 2011 at 11:49 | comment | added | Martin Brandenburg | mathoverflow.net/faq#whatnot | |
| Nov 8, 2011 at 11:37 | answer | added | Gerry Myerson | timeline score: 2 | |
| Nov 8, 2011 at 11:35 | history | edited | Gerry Myerson | CC BY-SA 3.0 |
added 3 characters in body; edited title
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| Nov 8, 2011 at 11:28 | history | asked | david | CC BY-SA 3.0 |