Exercise 10.3.10 (Diagrams)

Question

What do the diagrams of Example 10.3.9 look like for $p^8$ in place of $p^{12}$?
What about $p^{432}$?

Hassaan Naeem · Accepted Answer

extbf{Solution:}
\\
$p^8$:
\begin{equation*}
    \begin{tikzcd}
        \langle 	heta^{8} angle \cong C_1 \cong 1  \arrow[d, dash] & (	ext{order 1}) \
        \langle 	heta^{4} angle \cong C_2 \arrow[d, dash] & (	ext{order 2}) \
        \langle 	heta^{2} angle \cong C_4 \arrow[d, dash] & (	ext{order 4}) \
        G = \langle 	heta angle \cong C_{8} & (	ext{order 8}) \
    \end{tikzcd}
\end{equation*}
\begin{equation*}
    \begin{tikzcd}
        \mathbb{F}_{p^{8}} \arrow[d, dash]& (	ext{degree 1}) \
        \mathbb{F}_{p^4} \arrow[d, dash]& (	ext{degree 2}) \
        \mathbb{F}_{p^2} \arrow[d, dash]& (	ext{degree 4}) \
        \mathbb{F}_{p}& (	ext{degree 8})\
    \end{tikzcd}
\end{equation*}
\\
For $p^{432}$, follow same inutition. 432 has 20 divisors. Hence the diagram it quite large, hence I am not bothered
to draw it out.

Richard Ganaye · Answer

ewcommand{\Q}{\mathbb{Q}} ewcommand{\Z}{\mathbb{Z}} ewcommand{\N}{\mathbb{N}} ewcommand{\R}{\mathbb{R}} ewcommand{\C}{\mathbb{C}} ewcommand{\F}{\mathbb{F}} ewcommand{\U}{\mathbb{U}} ewcommand{ e}{\,\mathrm{Re}\,} ewcommand{\im}{\,\mathrm{Im}\,} ewcommand{\ord}{\mathrm{ord}} ewcommand{\Gal}{\mathrm{Gal}} \begin{proof} Galois correspondence for the subfields of $\F_{p^8}$: \medskip \begin{center} \begin{tikzpicture} ode (M) at (4,6) {$\F_{p^8}$} ; ode (FH1) at (4,4) {$\F_{p^4}$} ; ode (FH2) at (4,2) {$\F_{p^2}$} ; ode (K) at (4,0) {$\F_p$} ; \draw [->](K) edge (FH2) ; \draw [->](FH2) edge (FH1) ; \draw [->](FH1) edge (M) ; ode(1) at (8,6) {$\{e\}$} ; ode (H1) at (8,4) {$ \langle heta^4 angle \simeq C_2$} ; ode (H2) at (8,2) {$ \langle heta^2 angle \simeq C_4$} ; ode (G) at (8,0) {$\mathrm{Gal}(\F_{p^8}:\F_p) = \langle heta angle$} ; \draw [<-] (G) edge (H2) ; \draw [<-] (H2) edge (H1) ; \draw [<-] (H1) edge (1) ; \end{tikzpicture} \end{center} Subfields of $\F_{p^{432}}$: \medskip \begin{center} \begin{tikzpicture} ode (1) at (0,0) {$\F_p$} ; ode (2) at (-2, 1) {$\F_{p^2}$} ; ode (4) at (-4, 2) {$\F_{p^4}$} ; ode (8) at (-6, 3) {$\F_{p^8}$} ; ode (16) at (-8, 4) {$\F_{p^{16}}$} ; ode (3) at (2,2) {$\F_{p^{3}}$} ; ode (6) at (0,3) {$\F_{p^{6}}$} ; ode (12) at (-2,4) {$\F_{p^{12}}$} ; ode (24) at (-4,5) {$\F_{p^{24}}$} ; ode (48) at (-6,6) {$\F_{p^{48}}$} ; ode (9) at (4,4) {$\F_{p^{9}}$} ; ode (18) at (2,5) {$\F_{p^{18}}$} ; ode (36) at (0,6) {$\F_{p^{36}}$} ; ode (72) at (-2,7) {$\F_{p^{72}}$} ; ode (144) at (-4,8) {$\F_{p^{144}}$} ; ode (27) at (6,6) {$\F_{p^{27}}$} ; ode (54) at (4,7) {$\F_{p^{54}}$} ; ode (108) at (2,8) {$\F_{p^{108}}$} ; ode (216) at (0,9) {$\F_{p^{216}}$} ; ode (432) at (-2,10) {$\F_{p^{432}}$} ; \draw (1) edge (2) ; \draw (2) edge (4) ; \draw (4) edge (8) ; \draw (8) edge (16) ; \draw (3) edge (6) ; \draw (6) edge (12) ; \draw (12) edge (24) ; \draw (24) edge (48) ; \draw (9) edge (18) ; \draw (18) edge (36) ; \draw (36) edge (72) ; \draw (72) edge (144) ; \draw (27) edge (54) ; \draw (54) edge (108) ; \draw (108) edge (216) ; \draw (216) edge (432) ; \draw (1) edge (3) ; \draw (3) edge (9) ; \draw (9) edge (27) ; \draw (2) edge (6) ; \draw (6) edge (18) ; \draw (18) edge (54) ; \draw (4) edge (12) ; \draw (12) edge (36) ; \draw (36) edge (108) ; \draw (8) edge (24) ; \draw (24) edge (72) ; \draw (72) edge (216) ; \draw (16) edge (48) ; \draw (48) edge (144) ; \draw (144) edge (432) ; \end{tikzpicture} \end{center} The corresponding Galois groups: \medskip \begin{center} \begin{tikzpicture} ode (1) at (0,0) {$\langle heta angle$} ; ode (2) at (-2, 1) {$\langle heta^2 angle$} ; ode (4) at (-4, 2) {$\langle heta^4 angle$} ; ode (8) at (-6, 3) {$\langle heta^8 angle$} ; ode (16) at (-8, 4) {$\langle heta^{16} angle$} ; ode (3) at (2,2) {$\langle heta^{3} angle$} ; ode (6) at (0,3) {$\langle heta^{6} angle$} ; ode (12) at (-2,4) {$\langle heta^{12} angle$} ; ode (24) at (-4,5) {$\langle heta^{24} angle$} ; ode (48) at (-6,6) {$\langle heta^{48} angle$} ; ode (9) at (4,4) {$\langle heta^{9} angle$} ; ode (18) at (2,5) {$\langle heta^{18} angle$} ; ode (36) at (0,6) {$\langle heta^{36} angle$} ; ode (72) at (-2,7) {$\langle heta^{72} angle$} ; ode (144) at (-4,8) {$\langle heta^{144} angle$} ; ode (27) at (6,6) {$\langle heta^{27} angle$} ; ode (54) at (4,7) {$\langle heta^{54} angle$} ; ode (108) at (2,8) {$\langle heta^{108} angle$} ; ode (216) at (0,9) {$\langle heta^{216} angle$} ; ode (432) at (-2,10) {$\langle heta^{432} angle $} ; \draw (1) edge (2) ; \draw (2) edge (4) ; \draw (4) edge (8) ; \draw (8) edge (16) ; \draw (3) edge (6) ; \draw (6) edge (12) ; \draw (12) edge (24) ; \draw (24) edge (48) ; \draw (9) edge (18) ; \draw (18) edge (36) ; \draw (36) edge (72) ; \draw (72) edge (144) ; \draw (27) edge (54) ; \draw (54) edge (108) ; \draw (108) edge (216) ; \draw (216) edge (432) ; \draw (1) edge (3) ; \draw (3) edge (9) ; \draw (9) edge (27) ; \draw (2) edge (6) ; \draw (6) edge (18) ; \draw (18) edge (54) ; \draw (4) edge (12) ; \draw (12) edge (36) ; \draw (36) edge (108) ; \draw (8) edge (24) ; \draw (24) edge (72) ; \draw (72) edge (216) ; \draw (16) edge (48) ; \draw (48) edge (144) ; \draw (144) edge (432) ; \end{tikzpicture} \end{center} where $\langle heta^{432} angle = \{e\}, \langle heta angle = \Gal(\F_{p^{432}} : \F_p)$, and more generally $\langle heta^k angle = \Gal(\F_{p^{432}} : \F_{p^k}) \simeq C_{432/k}$. \end{proof}

Exercise 10.3.10 (Diagrams)

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Exercise 10.3.10 (Diagrams)

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