From 6987757d3bffa9c646a9597560cdf34e54b2983d Mon Sep 17 00:00:00 2001 From: user202729 <25191436+user202729@users.noreply.github.com> Date: Tue, 14 Nov 2023 20:54:51 +0000 Subject: [PATCH] Some changes (#209) * Clean up some labels and refs * Give some intuition/motivation for the Frobenius element and decomposition group * Fix the proof of Frobenius map restriction * Update frobenius.tex --------- Co-authored-by: Evan Chen --- tex/H113/sylow.tex | 2 +- tex/alg-NT/finite-field.tex | 1 + tex/alg-NT/frobenius.tex | 32 ++++++++++++++++---------------- tex/alg-NT/ramification.tex | 15 ++++++++++----- tex/frontmatter/digraph.tex | 2 +- tex/linalg/fourier.tex | 2 +- tex/measure/pontryagin.tex | 2 +- 7 files changed, 31 insertions(+), 25 deletions(-) diff --git a/tex/H113/sylow.tex b/tex/H113/sylow.tex index bd67b47f..b66d0e82 100644 --- a/tex/H113/sylow.tex +++ b/tex/H113/sylow.tex @@ -1,5 +1,5 @@ \chapter{Find all groups} -\label{chapter:sylow} +\label{ch:sylow} The following problem will hopefully never be proposed at the IMO. \begin{quote} Let $n$ be a positive integer and let $S = \left\{ 1,\dots,n \right\}$. diff --git a/tex/alg-NT/finite-field.tex b/tex/alg-NT/finite-field.tex index e7b8fa6f..814f1579 100644 --- a/tex/alg-NT/finite-field.tex +++ b/tex/alg-NT/finite-field.tex @@ -182,6 +182,7 @@ \section{The Galois theory of finite fields} \[ \sigma_p(x) = x^p \] is an automorphism, and moreover fixes $\FF_p$. \end{theorem} +This is called the Frobenius automorphism, and will re-appear later on in \Cref{ch:frobenius-element}. \begin{proof} It's a homomorphism since it fixes $1$, respects multiplication, diff --git a/tex/alg-NT/frobenius.tex b/tex/alg-NT/frobenius.tex index 910b9c78..9daa7026 100644 --- a/tex/alg-NT/frobenius.tex +++ b/tex/alg-NT/frobenius.tex @@ -1,4 +1,5 @@ \chapter{The Frobenius element} +\label{ch:frobenius-element} Throughout this chapter $K/\QQ$ is a Galois extension with Galois group $G$, $p$ is an \emph{unramified} rational prime in $K$, and $\kp$ is a prime above it. Picture: @@ -12,6 +13,12 @@ \chapter{The Frobenius element} \QQ & \supset & \ZZ & (p) & \FF_p \end{tikzcd} \end{center} + +We recall that the $p$-th power map $\sigma \colon \FF_{p^f} \to \FF_{p^f}$ is an automorphism, and it's called the Frobenius map on $\FF_{p^f}$. +We can try to extend this map to a $K \to K$ map by $\sigma(x) = x^p$, unfortunately this doesn't make it a field automorphism. + +Surprisingly, it is nevertheless possible to extend this to some field automorphism $\sigma \in \Gal(K/\QQ)$. + If $p$ is unramified, then one can show there is a unique $\sigma \in \Gal(K/\QQ)$ such that $\sigma(\alpha) \equiv \alpha^p \pmod{\kp}$ for every prime $p$. @@ -24,7 +31,7 @@ \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 $\alpha \in \OO_K$, \[ \Frob_\kp(\alpha) \equiv \alpha^{p} \pmod{\kp}. \] It is called the \vocab{Frobenius element} at $\kp$, and has order $f$. \end{theorem} @@ -338,6 +345,8 @@ \section{Frobenius elements behave well with restriction} \Frob_{\kP} \colon L \to L \] and want to know how these are related. +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. + \begin{theorem} [Restrictions of Frobenius elements] Assume $L/\QQ$ and $K/\QQ$ are both Galois. @@ -346,22 +355,13 @@ \section{Frobenius elements behave well with restriction} i.e.\ for every $\alpha \in K$, \[ \Frob_\kp(\alpha) = \Frob_{\kP}(\alpha). \] \end{theorem} -%\begin{proof} -% We know -% \[ \Frob_{\kP}(\alpha) \equiv \alpha^p \pmod{\kP} -% \quad \forall \alpha \in \OO_L \] -% from the definition. -% \begin{ques} -% Deduce that -% \[ \Frob_{\kP}(\alpha) \equiv \alpha^p \pmod{\kp} -% \quad \forall \alpha \in \OO_K. \] -% (This is weaker than the previous statement in two ways!) -% \end{ques} -% Thus $\Frob_{\kP}$ restricted to $\OO_K$ satisfies the -% characterizing property of $\Frob_\kp$. -%\end{proof} \begin{proof} - TODO: Broken proof. Needs repair. + 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. + + 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 $\OO_K / \kp \to \OO_L / \kP$. + + Since a field homomorphism is injective, $\Frob_{\kP}$ induces the power-of-$p$ map on $\OO_L / \kP$, and everything is commutative, the theorem follows. \end{proof} In short, the point of this section is that \begin{moral} diff --git a/tex/alg-NT/ramification.tex b/tex/alg-NT/ramification.tex index bc6b6010..8befa9bc 100644 --- a/tex/alg-NT/ramification.tex +++ b/tex/alg-NT/ramification.tex @@ -262,22 +262,27 @@ \section{(Optional) Decomposition and inertia groups} How are $\Gal\left( (\OO_K/\kp) / \FF_p \right)$ and $\Gal(K/\QQ)$ related? \end{quote} -Absurdly enough, there is an explicit answer: -\textbf{it's just the stabilizer of $\kp$, at least when -$p$ is unramified}. + +First, every $\sigma \in \Gal(K/\QQ)$ induces an automorphism of $\OO_K$, which induces a map +$\OO_K \to \OO_K/\kp$ by +\[ \alpha \mapsto \sigma(\alpha) \pmod\kp. \] +For this to induce a map in $\Gal\left( (\OO_K/\kp) / \FF_p \right)$, it's necessary that $\sigma(\kp) \subseteq \kp$. So, we consider the subset of automorphisms that fixes $\kp$: \begin{definition} Let $D_\kp \subseteq \Gal(K/\QQ)$ be the stabilizer of $\kp$, that is \[ D_\kp \defeq \left\{ \sigma \in \Gal(K/\QQ) \mid \sigma\kp = \kp \right\}. \] We say $D_\kp$ is the \vocab{decomposition group} of $\kp$. \end{definition} -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 $\Norm(\kp)$. + 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. +Surprisingly, every element of $\Gal\left( (\OO_K/\kp) / \FF_p \right)$ arises this way from some field automorphism of $K$. + \begin{theorem}[Decomposition group and Galois group] \label{thm:decomposition} Define $\theta$ as above. Then diff --git a/tex/frontmatter/digraph.tex b/tex/frontmatter/digraph.tex index 822ffbec..34d2e005 100644 --- a/tex/frontmatter/digraph.tex +++ b/tex/frontmatter/digraph.tex @@ -13,7 +13,7 @@ \reqhalfcourse 55,45:{Ch 2,6-8}{\hyperref[part:basictop]{Topology}}{} \halfcourse 33,45:{Ch 9-15,18}{\hyperref[part:linalg]{Lin Alg}}{} \halfcourse 5,35:{Ch 16}{\hyperref[part:groups]{Grp Act}}{} -\halfcourse 5,24:{Ch 17}{\hyperref[chapter:sylow]{Grp Classif}}{} +\halfcourse 5,24:{Ch 17}{\hyperref[ch:sylow]{Grp Classif}}{} \halfcourse 30,35:{Ch 19-22}{\hyperref[part:repth]{Rep Th}}{} \halfcourse 45,43:{Ch 23-25}{\hyperref[part:quantum]{Quantum}}{} \halfcourse 64,38:{Ch 26-30}{\hyperref[part:calc]{Calc}}{} diff --git a/tex/linalg/fourier.tex b/tex/linalg/fourier.tex index 1387c259..0ddfd652 100644 --- a/tex/linalg/fourier.tex +++ b/tex/linalg/fourier.tex @@ -344,7 +344,7 @@ \section{Summary, and another teaser} \section{Parseval and friends} Here is a fun section in which you get to learn a lot of big names quickly. Basically, we can take each of the three results -from Proposition~\ref{prop:orthonormal}, +from \Cref{prop:orthonormal}, translate it into the context of our Fourier analysis (for which we have an orthonormal basis of the Hilbert space), and get a big-name result. diff --git a/tex/measure/pontryagin.tex b/tex/measure/pontryagin.tex index f8457df7..be7c7fd8 100644 --- a/tex/measure/pontryagin.tex +++ b/tex/measure/pontryagin.tex @@ -181,7 +181,7 @@ \section{The orthonormal basis in the compact case} \end{itemize} \section{The Fourier transform of the non-compact case} -If $G$ is LCA but not compact, then Theorem~\ref{thm:god} becomes false. +If $G$ is LCA but not compact, then \Cref{thm:god} becomes false. On the other hand, it's still possible to define $\wh G$. We can then try to write the Fourier coefficients anyways: let \[ \wh f(\xi) = \int_G f \cdot \ol{e_\xi} \; d\mu \]