Timeline for Factorizing polynomials in $\mathbf{Z}[[x]]$
Current License: CC BY-SA 2.5
10 events
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| Feb 24, 2011 at 20:23 | comment | added | Pace Nielsen | I changed my answer. Hopefully I understood the question enough to write something coherent. | |
| Feb 24, 2011 at 20:20 | history | edited | Pace Nielsen | CC BY-SA 2.5 |
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| Feb 24, 2011 at 20:11 | history | edited | Pace Nielsen | CC BY-SA 2.5 |
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| Feb 24, 2011 at 17:55 | history | edited | Pace Nielsen | CC BY-SA 2.5 |
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| Feb 24, 2011 at 2:13 | comment | added | Tommaso Centeleghe | I do not understand your counterexamples. I think that in both cases, once you fix a root of your polynomial, you have only one place $\nu$ with the property above. Which, if my question has a positive answer, would mean that $f(x)$ stays irreducible in the power series ring. Which is the case in both your examples! | |
| Feb 23, 2011 at 17:23 | history | edited | Pace Nielsen | CC BY-SA 2.5 |
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| Feb 23, 2011 at 17:11 | history | edited | Pace Nielsen | CC BY-SA 2.5 |
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| Feb 23, 2011 at 0:46 | comment | added | Tommaso Centeleghe | I think Kevin is right. Notice more generally if the root $\alpha$ of an irreducible polynomial $f(x)$ has norm equal to one (i.e. leading term = constant term), then for every p-adic place $\nu$ of $K=Q(\alpha)$ for which $\alpha$ is not an integer there must be another p-adic place (same p!) $\nu'$ so that $\alpha$ belongs to the maximal ideal of the valuation ring of $\nu'$. This implies that if there is no place $\nu$ for which $\alpha$ belongs to the maximal ideal defined by $\nu$, then $\alpha$ is a unit and so is $f(x)$. | |
| Feb 22, 2011 at 23:32 | comment | added | Kevin Ventullo | Are you sure there are no such places $\nu$? If I haven't made a mistake, I think that 2 splits into $\mathfrak{p}\tilde{\mathfrak{p}}$ and that we can label the roots $\alpha$ and $\tilde{\alpha}$ such that $v_\mathfrak{p}(\alpha)$=$v_{\tilde{\mathfrak{p}}}(\tilde{\alpha})=1$ and $v_\mathfrak{p}(\tilde{\alpha})=v_{\tilde{\mathfrak{p}}}(\alpha)=-1$. | |
| Feb 22, 2011 at 19:43 | history | answered | Pace Nielsen | CC BY-SA 2.5 |