Timeline for Why are most coefficients of these minimal polynomials divisible by $p$?
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Sep 18, 2015 at 12:01 | vote | accept | Wolfgang | ||
| Sep 18, 2015 at 11:02 | comment | added | Joe Silverman | @Wolfgang Let $u$ and $v$ be algebraic numbers. Let $u_1,\ldots,u_n$ be the Galois conjugates of $u$, and let $v_1,\ldots,v_m$ be the Galois conjugates of $v$. Then $\{u_iv_j : 1\le i\le n,1\le j\le m\}$ is a complete set of Galois conjugates of $uv$, although there may, of course, be repeated values. This is clear, since $\sigma(uv)=\sigma(u)\sigma(v)$ for any $\sigma\in\text{Gal}(\overline{\mathbb Q}/\mathbb Q)$. So in the OP's case, the conjugates of $(1-\zeta^n)a$ all have the form $(1-\zeta^{nj})b$, where $b$ is some Galois conjugate of $a$. | |
| Sep 18, 2015 at 10:58 | comment | added | Wolfgang | I am still trying to understand the crucial "The minimal polynomial of $(1+\zeta^n)a$ is a product of terms of the form $X-(1+\zeta^{nj})\alpha_{ij}$". I believe you are right, but how do you know that the Galois conjugates of ζ are sufficient to "capture it all"? And what role does the p-adic valuation of a play here? | |
| Sep 17, 2015 at 23:47 | comment | added | Joe Silverman | @GerhardPaseman I guess I'm just too used to taking $\pi$ to be 1 minus a $p$'th root of unity, with the norm of $\pi$ generating the ideal $p\mathbb Z$. But you're probably right that for this problem it would be easier to just use $1+\zeta$ with $\zeta$ a primitive $2p$'th root of unity. | |
| Sep 17, 2015 at 20:37 | comment | added | Gerhard Paseman | I find the conflict in notation unfortunate. If instead you just used $\zeta$, and reminded people that $\zeta = -1 \bmod (1 + \zeta)$, I would find your presentation much easier to read. Gerhard "Also Needs Fewer Greek Letters" Paseman, 2015.09.17 | |
| Sep 17, 2015 at 19:24 | history | edited | Joe Silverman | CC BY-SA 3.0 |
added 89 characters in body
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| Sep 17, 2015 at 18:24 | history | answered | Joe Silverman | CC BY-SA 3.0 |