Exercise 10.1.4

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}}

This exercise covers the details omitted in the proof of Theorem 10.1.6.
\be
\item[(a)] Suppose that a line $l_1$ goes through distinct points $\alpha_1 = u_1+iv_1$ and $\beta_1 = u_2+iv_2$, where $u_1,v_1,u_2,v_2$ lie in a subfield $F \subset \R$. Prove that $l_1$ is defined by an equation of the form $a_1x +b_1y = c_1$ where $a_1,b_1,c_1 \in F$.
\item[(b)] Suppose that $\alpha_2 
e \beta_2$ and $\gamma_2$ are complex numbers whose real and imaginary parts lie in a subfield $F\subset \R$. Prove that the circle $C$ with center $\gamma_2$ and radius $|\alpha_2-\beta_2|$ has an equation of the form (10.3) with $a_2,b_2,c_2 \in F$.
\item[(c)] In the proof of Theorem 10.1.6, we considered the equations (10.2) and (10.3) when $a_1
e 0$. Explain what happens when $a_1 = 0$ in (10.2).
\ee

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}
\be
\item[(a)]
 Let $M = (x,y),A = (u_1,v_1),B = (u_2,v_2)$ be the points of $\R^2$ with affixes $z,\alpha_1,\beta_1$, where $\alpha_1
e \beta_1$.
Then 
\begin{align*}
M \in (AB) &\iff \det(\overrightarrow{AM}, \overrightarrow{AB}) = 0\
&\iff
\left|
\begin{array}{ccc}
  x-u_1& u_2-u_1    \
  y-v_1& v_2-v_1      
\end{array}
ight|
=0\
&\iff (v_2-v_1)x - (u_2-u_1)y -u_1(v_2-v_1) +v_1 (u_2-u_1) = 0\
&\iff a_1x+b_1y = c_1
\end{align*}
where $a_1 = v_2-v_1, b_1 = -(u_2-u_1), c_1 = u_1(v_2-v_1) - v_1(u_2-u_1) \in F$.
\item[(b)] Let $z = x+iy, \gamma_2 = u+iv, \alpha_2 - \beta_2 = a +ib$, where $u,v,a,b \in F$. Then
\begin{align*}
z \in C &\iff |z-\gamma_2|^2 = |\alpha_2-\beta_2|^2\
&\iff(x-u)^2 + (y-v)^2 = a^2+b^2\
&\iff x^2+y^2+a_2x+b_2y+c_2 = 0,
\end{align*}
where $a_2 = -2u,b_2 = -2v ,c_2 = -(a^2+b^2) \in F$.
\item[(c)] If $a_1 = 0$, then for all $z =x+iy \in l$, $y=c_1/b_1 = u$, where $u = c_1/b_1 \in F=F_n$. Substituting $y = u$ into (10.3) gives the quadratic equation
$$x^2+a_2x+u^2+b_2u+c_2.$$
Therefore $x$ lies in a quadratic extension $F_{n+1}$ of $F_n$ (and $y \in F_n \subset F_{n+1}$).
\ee
\end{proof}

Exercise 10.1.4

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Exercise 10.1.4

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