Some changes #209
Some changes #209
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| @@ -24,7 +30,7 @@ \section{Frobenius elements} | |||
| Assume $K/\QQ$ is Galois with Galois group $G$. | |||
| Let $p$ be a rational prime unramified in $K$, and $\kp$ a prime above it. | |||
| There is a \emph{unique} element $\Frob_\kp \in G$ | |||
| with the property that | |||
| with the property that, for all $\alpha \in \OO_K$, | |||
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For clarity. (actually thinking about it, is "mod 𝔭" well-defined over the whole field K? If K = ℚ[i] then obviously yes but...?)
Edit: should be yes, consider the additive structure of K which is Abelian, so 𝔭 is a (normal) subgroup and the quotient group K/𝔭 is well-defined. (although it's not well-defined as a ring or a K-module)
| @@ -338,6 +344,8 @@ \section{Frobenius elements behave well with restriction} | |||
| \Frob_{\kP} \colon L \to L \] | |||
| and want to know how these are related. | |||
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| Both maps $\Frob_{\kP}$ and $\Frob_{\kp}$ induce the power-of-$p$ map in the corresponding quotient field, hence we would expect them to be naturally the same. | |||
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thus we would expect the proof of consist of only "typecasting".
Perhaps that's what I mean, but my ability to write isn't that good.
| First, $K/\QQ$ is normal, so $\Frob_{\kP}$ fixes the image of $K$, that is, | ||
| $\Frob_{\kP} \restrict{K} \in \Gal(K/\QQ)$ is well-defined. | ||
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| We have the natural map $\phi \colon \OO_K \to \OO_L \to \OO_L/\kP$, and the quotient map $q\colon \OO_K \to \OO_K / \kp$. Since $\kP \subseteq \kP$, then $\phi$ factors through $q$ to give a natural field homomorphism $\OO_K / \kp \to \OO_L / \kP$. |
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Seem like an interesting side result, but I don't know if this is useful in other ways.
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Only after looking at it compiled I realize I made a mistake, it's supposed to say 𝔭 ⊆ 𝔓. (but the more correct thing to say would be 𝔭 ⊆ 𝔓 ∩ 𝒪ₖ ⊆ ker φ, which is the reason why φ factors through q)
| Absurdly enough, there is an explicit answer: | ||
| \textbf{it's just the stabilizer of $\kp$, at least when | ||
| $p$ is unramified}. | ||
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("It's natural to consider the stabilizer, not absurd!")
In fact you somewhat derived the result right below (explain how we should consider the stabilizer), I mostly just reorder it to motivate it better.
| Then, every $\sigma \in D_\kp$ induces an automorphism of $\OO_K / \kp$ by | ||
| \[ \alpha \mapsto \sigma(\alpha) \pmod\kp. \] | ||
| Note that this definition is in fact equivalent to the set of $\sigma$ such that $\sigma(\kp) \subseteq \kp$, | ||
| because a field isomorphism fixes the ideal norm $\Norm(\kp)$. |
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Is there a better proof of this fact?
| So there's a natural map | ||
| \[ D_\kp \taking\theta \Gal\left( (\OO_K/\kp) / \FF_p \right) \] | ||
| by declaring $\theta(\sigma)$ to just be ``$\sigma \pmod \kp$''. | ||
| The fact that $\sigma \in D_\kp$ (i.e.\ $\sigma$ fixes $\kp$) | ||
| ensures this map is well-defined. | ||
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| Surprisingly, every element of $\Gal\left( (\OO_K/\kp) / \FF_p \right)$ arises this way from some field automorphism of $K$. |
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Maybe worth saying something about what the kernel mean, but I haven't gotten the intuition yet.
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Looks nice, thanks much! |
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