I'm trying to show that the constructible elements form a field.
My definition of constructible is that a point in the plane $P=(a,b)\in \mathbb{R}^2$ is constructible if there exists a set of points $ \{P_0, P_1, · · · , P_n=P\}$ containing $P_0=(0,0)$ and $P_1=(1,0)$ such that $P_i$ is either an intersection of two lines or an intersection of one line and one circles or an intersection of two circle obtained from the set $ \{P_0, P_1, · · · , P_{i-1}\}$.
I was able to show that if $P,P'$ are constructible, then so are $P+P'$ and $-P$ but I find it harder to show that $P^{-1}$ and $P\cdot P'$ are also constructible. For $P^{-1}$ I tried to write it like $P^{-1}=\frac{\bar{P}}{||P||}$ where $\bar{P}=(a,-b)$ ( the complex conjugute) but I couldn't go any further.
Any hints on how to show that $P^{-1}$ and $P \cdot P'$ are also constructible?
To demonstrate that the constructible elements form a field, we need to establish closure under subtraction, multiplication, and inversion for any two constructible points $P = (a, b)$ and $P' = (a', b')$. We have already shown closure under addition and negation.
$\textbf{Closure under Subtraction:}$ Let $P - P'$ be the difference of two constructible points. Construct the line passing through $P'$ and $P$, and find the intersection points with the original constructible set. These intersection points satisfy the conditions for constructibility, thus demonstrating closure under subtraction.
$\textbf{Closure under Multiplication:}$ For $(P \cdot P')$, construct a line through $(P')$ and $(1, 0)$, which intersects the unit circle at a point $Q$. Construct the line through $Q$ and $P$, and the intersection with the x-axis gives $P \cdot P'$, proving closure under multiplication.
$\textbf{Closure under Inversion:}$ For $P^{-1}$, construct the point $(a^2 + b^2, 0)$ by drawing a circle with radius $\sqrt{a^2 + b^2}$ centered at the origin. The vertical line through this point intersects the unit circle at $P^{-1}$, establishing closure under inversion.
These constructions demonstrate that the set of constructible points forms a field, satisfying the necessary conditions for closure under basic field operations.