Exercise 8.1.5 (Galois correspondence diagram)

Question

Draw a diagram like Figure 8.1 for the second half of
Lemma 8.1.2(i).

Hassaan Naeem · Accepted Answer

extbf{Solution:}
\begin{equation*}
\begin{tikzcd}
    & 1 \arrow[d, dash] & M \arrow[d, dash]\
    & H_1 \arrow[d, dash] & Fix(H_1) \arrow[d, dash]\
    & H_2 \arrow[d, dash] & Fix(H_2) \arrow[d, dash]\
    & Gal(M:K) & K
\end{tikzcd}
\end{equation*}

Richard Ganaye · Answer

\begin{proof} $ H_1 \subseteq H_2 \Rightarrow \mathrm{Fix}(H_1) \supseteq \mathrm{Fix}(H_2).$ \bigskip \begin{center} \begin{tikzpicture} ode(1) at (0,6) {$1$} ; ode (H1) at (0,4) {$H_1$} ; ode (H2) at (0,2) {$H_2$} ; ode (G) at (0,0) {$\mathrm{Gal}(M:K)$} ; \draw (G) edge (H2) ; \draw (H2) edge (H1) ; \draw (H1) edge (1) ; ode (M) at (4,6) {$M$} ; ode (FH1) at (4,4) {$\mathrm{Fix}(H_1)$} ; ode (FH2) at (4,2) {$\mathrm{Fix}(H_2)$} ; ode (K) at (4,0) {$K$} ; \draw (K) edge (FH2) ; \draw (FH2) edge (FH1) ; \draw (FH1) edge (M) ; \end{tikzpicture} \end{center} \medskip If we use arrows to see the inclusions, this gives \medskip \begin{center} \begin{tikzpicture} ode(1) at (0,6) {$1$} ; ode (H1) at (0,4) {$H_1$} ; ode (H2) at (0,2) {$H_2$} ; ode (G) at (0,0) {$\mathrm{Gal}(M:K)$} ; \draw [<-] (G) edge (H2) ; \draw [<-] (H2) edge (H1) ; \draw [<-] (H1) edge (1) ; ode [<-](M) at (4,6) {$M$} ; ode (FH1) at (4,4) {$\mathrm{Fix}(H_1)$} ; ode (FH2) at (4,2) {$\mathrm{Fix}(H_2)$} ; ode (K) at (4,0) {$K$} ; \draw [->](K) edge (FH2) ; \draw [->](FH2) edge (FH1) ; \draw [->](FH1) edge (M) ; \end{tikzpicture} \end{center} \end{proof}

Exercise 8.1.5 (Galois correspondence diagram)

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Exercise 8.1.5 (Galois correspondence diagram)

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