Exercise 10.1.1

Question

\def\imp{\Rightarrow}
\def\gcro{\mbox{[\hspace{-.15em}[}}% intervalles d'entiers 
\def\dcro{\mbox{]\hspace{-.15em}]}}

ewcommand{\be} {\begin{enumerate}}

ewcommand{\ee} {\end{enumerate}}

ewcommand{\deb}{\begin{eqnarray*}}

ewcommand{\fin}{\end{eqnarray*}}

ewcommand{\ssi} {si et seulement si }

ewcommand{\D}{\mathrm{d}}

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}}

ewcommand{\legendre}[2]{\genfrac{(}{)}{}{}{#1}{#2}}

In part (a) of Example 10.1.2 we constructed the $x$-axis. In a similar way show that the $y$-axis is constructible. For each step in your construction be sure to say which of $C1,C2,P1,P2$, and $P3$ you are using.

Richard Ganaye · Accepted Answer

\def\imp{\Rightarrow}
\def\gcro{\mbox{[\hspace{-.15em}[}}% intervalles d'entiers 
\def\dcro{\mbox{]\hspace{-.15em}]}}

ewcommand{\be} {\begin{enumerate}}

ewcommand{\ee} {\end{enumerate}}

ewcommand{\deb}{\begin{eqnarray*}}

ewcommand{\fin}{\end{eqnarray*}}

ewcommand{\ssi} {si et seulement si }

ewcommand{\D}{\mathrm{d}}

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}}

ewcommand{\legendre}[2]{\genfrac{(}{)}{}{}{#1}{#2}}

\begin{proof}
Let $ \mathscr{C}$ be the set of constructible numbers in $\C$.

Write $C(\gamma,r)$ the circle with center $\gamma \in \C$ and radius $r$.

Starting from $ \{0,1\}$, we obtain by $C1$ the $x$-axis and by $P2$ the point $-1$ at the intersection of $C(0,1)$ with the $x$-axis, so
$$ \{-1,0,1\} \subset {\mathscr{C}}.$$
If $\alpha = -1, \beta = 1$, then $|\beta - \alpha| = 2$, so $C(-1,2)$ and $C(1,2)$ are constructible by $C2$.

Using $P3$, the intersection $C(-1,2) \cap C(1,2)=\{\delta, - \delta\}$ gives the point $\delta = i\sqrt{3}$. The $y$-axis that goes through $0$ and $\delta$ is so constructible by $C1$.

Example 10.1.2(b) shows then that $i \in \C$ is  constructible. 
\end{proof}

Exercise 10.1.1

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Exercise 10.1.1

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