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zn|t5gzU}#FUWSa{3f7!g;QYc0Y+oHM#yydmYR0rFq`Z8zqvmdF>z*Aiez1 zc9{c`n9j)+7WMYKxZ8lh%^wH{4+-aZ37vbaWx{PE-EEw-{?!*)F770)y*f_UtO;}3 zRQhI(08Ycm;#Fh5&(7(r0etqFOhcS38BPtKuy$raH%dMS64zNC1kaVx5n88V7`S~? zI;o{gaAAxyxOqvQYblPYW+tlxRJPs}rRRd$%RMIM{A<$UjfB1dPeO~Ql~d_izZsFU z=8#2pwBta1+4R-MS2F@TWB&!>LT|;E`oIcLyvce*qZILYDzecRGD>@GKL&l1ZZMOg zBa(E)sTcETJdbG5iO?OGQYCp8dWg%ygvuhCYB7i<;#^=~DjI_V*Y8u3!uZDRs-sXj g{@MItP`FI3)Q0eGo)#XtC2enP)6aWAK diff --git a/galois-theory-notes.tex b/galois-theory-notes.tex index 64fea4e..740342d 100644 --- a/galois-theory-notes.tex +++ b/galois-theory-notes.tex @@ -37,6 +37,21 @@ note = {\url{https://scipp.ucsc.edu/~haber/ph251/Dn_subgroups.pdf}}, url = {https://scipp.ucsc.edu/~haber/ph251/Dn_subgroups.pdf} } + +@misc{judson, + author={Thomas W. Judson}, + title={Abstract Algebra: theory and applications}, + year={1994}, + note={Available at \url{http://abstract.ups.edu/download.html} }, + pages={438} +} + +@misc{dummitfoote, + author={David S. Dummit and Richard M. Foote}, + title={Abstract Algebra (Third Edition)}, + year={2004}, + pages={945} +} \end{filecontents} \nocite{*} @@ -87,7 +102,8 @@ \tableofcontents -\section{Recap on the degree of field extensions} +\section{Galois Theory notes} +\subsection{Chapters 4-6} (Definitions, theorems, lemmas, corollaries and examples enumeration follows from Ian Stewart's book \cite{ianstewart}). \begin{defn}{4.10} @@ -176,6 +192,201 @@ [...] TODO: pending to add key parts up to Chapter 15. +\subsection{Detour: Isomorphism Theorems} +\begin{thm}{}(\emph{First Isomorphism Theorem}) \label{1stisothm} + + \begin{minipage}{0.75\textwidth} + If $\psi: G \longrightarrow H$ a group homomorphism, then $ker(\psi) \triangleleft G$.\\ + Let $\phi: G \longrightarrow G/ker(\psi)$ be the canonical homomorphism.\\ + Then $\exists$ unique isomorphism $\eta: G/ker(\psi) \longrightarrow \psi(G)$ such that $\psi = \eta \phi$.\\ + $\Longleftrightarrow$ ie. $G/ker(\psi) \cong \psi(G)$. + \end{minipage} +\hfill +\begin{minipage}{0.2\textwidth} + \begin{tikzpicture}[node distance=2.5cm, auto] + \node (G) {$G$}; + \node (H) [right of=G] {$H$}; + \node (GmodK) [below of=G, xshift=1.25cm] {$G/\ker(\psi)$}; + + \draw[->] (G) to node {$\psi$} (H); + \draw[->] (G) to node [swap] {$\phi$} (GmodK); + \draw[->] (GmodK) to node [swap] {$\eta$} (H); + \end{tikzpicture} +\end{minipage} +\end{thm} +\begin{proof} + \emph{(proof from Thomas W. Judson book "Abstract Algebra" \cite{judson})}\\ + + Let $K=ker(\psi)$. + Since + $$\eta: G/K \longrightarrow \psi(G)$$ + let + $$\eta: gK \longrightarrow \psi(g)$$ + ie. $\eta(gK)=\psi(g)$. + + \begin{enumerate}[i.] + \item show that $\eta$ is a \emph{well defined} map: + + if $g_1 K=g_2 K$, then for some $k \in K$, $g_1 k =g_2$, so + $$\eta(g_1K)=\psi(g_1) = \psi(g_1)\psi(k) = \psi(g_1 k) = \psi(g_2) = \eta(g_2 k)$$ + + Thus, $\eta$ does not depend on the choice of coset representatives, and + the map $\eta: G/ker(\psi) \longrightarrow \psi(G)$ is uniquely defined + since $\psi=\eta\phi$. + + \item show that $\eta$ is a homomorphism: + + Observe: + $$\eta(g_1 K g_2 K) = \eta(g_1 g_2 K) = \psi(g_1 g_2) = \psi(g_1) \psi(g_2) = \eta(g_1 K) \eta(g_2 K)$$ + $\Longrightarrow$ so $\eta$ is a homomorphism. + + \item show that $\eta$ is an isomorphism: + + Since each element of $H=\psi(G)$ has at least a preimage, then $\eta$ is \emph{surjective} (onto $\psi(G)$). + + Show that it is also \emph{injective} (onet-to-one): + + Suppose 2 different preimatges lead to the same image in $\psi(G)$, ie. + $\eta(g_1 K) = \eta(g_2 K)$ + + then, + $$\psi(g_1) = \psi(g_2)$$ + which implies $\psi(g_1^{-1} g_2) = e$, ie. $g_1^{-1} g_2 \in ker(\psi)$, + hence + $$g_1^{-1} g_2 K = K$$ + $$g_1 K = g_2 K$$ + so $\eta$ is injective. + \end{enumerate} + + Since $\eta$ is injective and surjective $\Longrightarrow$ $\eta$ is a bijective homomorphism,\\ + ie. $\eta$ is an \emph{isomorphism}. +\end{proof} + + +\begin{thm}{}(\emph{Second Isomorphism Theorem}) \label{2ndisothm} + Let $H \subseteq G$, $N \triangleleft G$. Then + \begin{enumerate}[i.] + \item $HN \subseteq G$ + \item $H \cap N \triangleleft H$ + \item $\frac{H}{H \cap N} \cong \frac{HN}{N}$ + \end{enumerate} +\end{thm} +\begin{proof} + \emph{(proof from Thomas W. Judson book "Abstract Algebra" \cite{judson})}\\ + + \begin{enumerate}[i.] + \item show $HN \subseteq G$:\\ + Note that $HN = \{ hn : h\in H, n\in N \}$. Let $h_1 n_1, h_2 n_2 \in HN$. + + Since $N$ normal $\Longrightarrow~ h_2^{-1} n_1 h_2 \in N$, so + $$(h_1 n_1)(h_2 n_2) = h_1 h_2 (h_2^{-1} n_1 h_2) \in HN$$ + + [Recall: since $N \triangleleft G$, $gN=Ng ~\forall g \in G$ $\Longrightarrow gn=n'g$ for some $n' \in N$.] + + To see that $(hn)^{-1} \in HN$:\\ + since $(hn)^{-1} = n^{-1} h^{-1} = h^{-1} (h n^{-1} h^{-1})$, thus $(hn)^{-1} \in HN$. + + Thus $HN \subseteq G$. + + + In fact, $$HN = \bigcup_{h \in H} hN$$ + + (TODO: diagram) + + \item show that $H \cap N \triangleleft H$:\\ + Let $h \in H,~ n \in H \cap N$ (recall: $H \cap N \subseteq H$).\\ + Then $h^{-1}nh \in H$ $\longleftarrow$ since $h^{-1}, n, h \in H$.\\ + Since $N \triangleleft G$, $h^{-1} n h \in N$.\\ + Therefore, $h^{-1}nh \in H \cap N$ $\Longrightarrow~ H \cap N \triangleleft H$ + + \item show that $\frac{H}{H \cap N} \cong \frac{HN}{N}$:\\ + Define a map + \begin{align*} + \phi: &H \longrightarrow \frac{HN}{N}\\ + \text{by}~ \phi: &h \longmapsto hN + \end{align*} + + $\phi$ is surjective (onto), since any coset $hnN=hN$ is the image of $h \in H$, ie. $\phi(h)$ + + $\phi$ is a homomorphism, since + $$\phi(h h') = h h' N = hN h'N = \phi(h)\phi(h')$$ + + By the First Isomorphism Theorem \ref{1stisothm}, + $$\frac{HN}{N} \cong \frac{H}{ker(\phi)}$$ + + and since + \begin{align*} + &ker(\phi) = \{ h \in H : h \in N\}\\ + &\text{then}~~ker(\phi) = H \cap N + \end{align*} + + so then, + $$\frac{HN}{N} = \phi(H) \cong \frac{H}{ker(\phi)} = \frac{H}{H \cap N}$$ + thus + $$\frac{HN}{N} \cong \frac{H}{H \cap N}$$ + + \end{enumerate} +\end{proof} + + +\begin{thm}{}(\emph{Third Isomorphism Theorem}) \label{2ndisothm}\\ + Let $H \subseteq K$ and $K \triangleleft G,~ H \triangleleft G$.\\ + Then + $\frac{K}{H} \triangleleft \frac{G}{H}$ + and + $$\frac{ G/H }{ K/H } \cong \frac{G}{K}$$ +\end{thm} +\begin{proof} + \emph{(proof from Dummit and Foote book ``Abstract Algebra" \cite{dummitfoote})}\\ + + Easy to see that $\frac{K}{H} \triangleleft \frac{G}{H}$. + + Define + \begin{align*} + \psi: &\frac{G}{H} \longrightarrow \frac{G}{K}\\ + \text{by}~ \psi: &gH \longmapsto gK + \end{align*} + + To show that $\psi$ is \emph{well defined}:\\ + suppose $g_1 H = g_2 H$, then $g_1 = g_2 h$ for some $h \in H$.\\ + Since $H \subseteq K \Longrightarrow~ h \in K$, hence $g_1 K = g_2 K$,\\ + ie. $\psi(g_1 H) = \psi(g_2 H)$, which shows that $\psi$ is well defined. + + Since $g \in G$ may be chosen arbitrarily in $G$, $\psi$ is a surjective homomorphism. + + + Finally, + \begin{align*} + ker(\psi) &= \{ gH \in \frac{G}{H} \mid \psi(gH)=1K \}\\ + &= \{ gH \in \frac{G}{H} \mid gK =1K \}\\ + &= \{ gH \in \frac{G}{H} \mid g \in K \}\\ + &= \frac{K}{H} + \end{align*} + + + By the First Isomorphism Theorem (\ref{1stisothm}), + + \begin{tikzpicture}[node distance=2.5cm, auto] + \node (GmodH) {$\frac{G}{H}$}; + \node (GmodK) [right of=GmodH] {$\frac{G}{K}$}; + \node (GmodHmodKer) [below of=G, xshift=1.25cm] {$\frac{G/H}{\ker(\psi)} = \frac{G/H}{K/H}$}; + + \draw[->] (GmodH) to node {$\psi$} (GmodK); + \draw[->] (GmodH) to node [swap] {$\phi$} (GmodHmodKer); + \draw[->] (GmodHmodKer) to node [swap] {$\eta$} (GmodK); + \end{tikzpicture} + + So, by + $$\eta: \frac{G/H}{K/H} \longrightarrow \frac{G}{K}$$ + since $\eta$ is + bijective (we know it by the First Isomorphism Theorem), $\eta$ it is the isomorphism: + $$\frac{ G/H }{ K/H } \cong \frac{G}{K}$$ + + +\end{proof} + +\subsection{Chapter 14} + \newpage @@ -296,7 +507,7 @@ $$t^{12}-1 = \Phi_1 \Phi_2 \Phi_3 \Phi_4 \Phi_6 \Phi_{12}$$ \begin{defn}{21.5} The polynomial $\Phi_d(t)$ defined by $$\Phi_n(t) = \prod_{a\in \mathbb{Z}_n,(a,n)=1} (t- \zeta^a)$$ - is the $n$-th \emph{cyclotomic polynomial} over \mathbb{C}. + is the $n$-th \emph{cyclotomic polynomial} over $\mathbb{C}$. \end{defn} \begin{cor}{21.6} @@ -305,14 +516,15 @@ $$t^{12}-1 = \Phi_1 \Phi_2 \Phi_3 \Phi_4 \Phi_6 \Phi_{12}$$ \begin{thm}{21.9} \begin{enumerate} - \item The Galois group $\Gamma(\mathbb{Q}(\zeta):\mathbb{Q})$ consists of the - $\mathbb{Q}$-automorphisms $\psi_j$ defined by - $$\psi_j(\zeta)=\zeta^j$$ - where $0 \leq j \leq n-1$ and $j$ is prime to $n$. - - \item $\Gamma(\mathbb{Q}(\zeta):\mathbb{Q}) \stackrel{iso}{\cong} \mathbb{Z}_n^*$, and is an abelian group. - \item its order is $\phi(n)$ - \item if $n$ is prime, $\mathbb{Z}_n^*$ is cyclic + \item The Galois group $\Gamma(\mathbb{Q}(\zeta):\mathbb{Q})$ consists of the + $\mathbb{Q}$-automorphisms $\psi_j$ defined by + $$\psi_j(\zeta)=\zeta^j$$ + where $0 \leq j \leq n-1$ and $j$ is prime to $n$. + + \item $\Gamma(\mathbb{Q}(\zeta):\mathbb{Q}) \stackrel{iso}{\cong} \mathbb{Z}_n^*$, and is an abelian group. + \item its order is $\phi(n)$ + \item if $n$ is prime, $\mathbb{Z}_n^*$ is cyclic + \end{enumerate} \end{thm}