Dependencies

A circle \(c\) is in \(\mathcal{C(M_{\infty })}\) if and only if \(c\) is in \(\mathcal{C(M_i)}\) for some \(i\).

For \(z \in \mathbb {C}\), if there is no \(m\) such that \([z:K_0] = 2^m\) then \(z \notin M_{\infty }\).

For \(z \in \mathbb {C}\), \(z \in M_{\infty }\) with \([K_0:\mathbb {Q}] = 2^n\), if there is no \(m\) such that \([z:\mathbb {Q}] = 2^m\) then \(z \notin M_{\infty }\).

If \(z_1, z_2 \in M_{\infty }\), then \(\frac{z_1 + z_2}{2} \in M_{\infty }\).

\(\imath\in M_{\infty }\).

For \(z_1, z_2 \in M_{\infty }\) it follows that \(z_1 \cdot z_2\) in \(M_{\infty }\).

For \(z_1, z_2 \in M_{\infty }\) it follows that \(z_1 - z_2 \in M_{\infty }\).

If \(z \in M_{\infty }\), then \(z^{-1}\) is in \(M_{\infty }\).

For \(M\subseteq \mathbb {C}\) with \(0,1 \in M\), \(K_0\) is conjugate closed.

A line \(l\) is in \(\mathcal{L(M_{\infty })}\) if and only if \(l\) is in \(\mathcal{L(M_i)}\) for some \(i\).

If \(z \in M_{\infty }\), then \(\sqrt{z}\) is in \(M_{\infty }\).

For \(z \in \mathbb {C}\). \(z\) is in \(M_{\infty }\) if and only if \(z.re, z.im \in M_{\infty }\).

A circle \(C := (c,r)\) is defined by a center \(c\) with \(c\in \mathbb {C}\), a radius \(r\in \mathbb {R}_{\ge 0}\) and the set:

We say two circles are equal if their underlying sets are equal.

We call a subset of \(\mathbb {C}\) conjugate closed if \(M= Conj(M)\).

For a Set \(M \subset \mathbb {C}\) we define the conjugate set of \(M\) as

The direction vector of a line \(l\) is the vector \(l.z_1 - l.z_2\).

Let \(\mathcal(M)\subseteq \mathbb {C}\) with \(0,1 \in \mathcal{M}\). Define:

A line \(L := (x,y)\) is defined by two points \(x,y\in \mathbb {C}\) with \(x\ne y\) and a set

We say two lines are equal if their generated set is equal.

Two lines \(l_1\) and \(l_2\) are parallel if they have the same slope, i.e. \(\exists r\in \mathbb {C}: l_1 = \{ x + r \mid x\in l_2\} \).

Two lines \(l_1\) and \(l_2\) are parallel’ if there exists a \(k\) such that \(l_1.z_1 - l_1.z_2 = k \cdot (l_2.z_1 - l_2.z_2)\).

A field \(F\) is called quadratic closed if for all \(x \in F\) there is a \(y \in F\) such that \(y^2 = x\).

We define operations that can be used to construct new points.

1. \((ILL)\) is the intersection of two different lines in \(\mathcal{L(M)}\).

2. \((ILC)\) is the intersection of a line in \(\mathcal{L(M)}\) and a circle in \(\mathcal{C(M)}\).

3. \((ICC)\) is the intersection of two different circles in \(\mathcal{C(M)}\).

\(ICL(\mathcal{M})\) is the union of all points that can be constructed using the operations \((ILL)\), \((ILC)\) and \((ICC)\) and \(\mathcal{M}\).

\(\mathcal{C(M)}\) is the set of all circles in \(\mathbb {C}\), with center in \(\mathcal{M}\) and radius of \(\mathcal{C}\) which is the distance of two points in \(\mathcal{M}\). As an equation this is:

We inductively define the chain

with \(\mathcal{M}_0 = \mathcal{M}\) and \(\mathcal{M}_{n+1} = ICL(\mathcal{M}_n)\)
and call \(\mathcal{M}_{\infty } = \bigcup _{n \in \mathbb {N}} \mathcal{M}_n\) the set of all constructable points.

\(\mathcal{L(M)}\) is the set of all real straight lines \(l\), with \(| l\cap \mathcal{M} |\ge 2\). As a set this is:

If \(0 \ne z = r\cdot e^{\imath\alpha } \in M_{\infty }\), then \(e^{\imath\alpha } \in M_{\infty }\).

If we have two circles \(C_1 = (c_1,r_1)\), \(C_2 = (c_2,r_2)\) and we have \(c_1 \ne c_2\) or \(r_1 \ne r_2 \) then the circles aren’t equal.

For \(z \in \mathbb {C}\), \(z \in M_{\infty }\) implies there exists a \(m\) such that \([z:K_0] = 2^m\).

Let \(K\) be an conjugation closed intermediate field of \(\mathbb {Q}\) and \(\mathbb {C}\) and \( M \subset \mathbb {C}\) be a subset with \(M = conj(M)\). Then \(K(M)\) is conjugate closed.

For a set \(M \subset \mathbb {C}\) it holds that \(Conj(Conj(M)) = M\).

\(M_{\infty }\) is conjugate closed.

For two sets \(M,N \subset \mathbb {C}\)

For a subfield \(F\) of \(\mathbb {C}\) the conjugate set \(Conj(F)\) is a subfield of \(\mathbb {C}\).

For \(M,N \subseteq \mathbb {C}\) with \(M \subseteq N\) it holds that \(Conj(M) \subseteq Conj(N)\).

Let \(L\) be a subfield of \(\mathbb {C}\), with \(L = conj(L)\), and \(z_1, z_2 \in L\). For \(r := \| z_1-z_2\| \) we get that \(r^2 \in L\).

Let \(L\) be a subfield of \(\mathbb {C}\), with \(L = conj(L)\). For \(i = 1,2 \) let \(z_i = x_i + \imath y_i \in L\) with \(z_1 \ne z_2\) and let \(r_i {\gt} 0\) in \(\mathbb {R}\) with \(r_i^2 \in L\). Define

Assume \(C_1 \cap C_2 \ne \emptyset \) and \(C_1 \ne C_2\). Then ther exists \(z_1, z_2 \in L\) such that

is a real line, and

For \(z \in C_1 \cap C_2\) there is a \(w \in L\) such that \(z \in L(\sqrt{w})\).

Let \(L\) be a subfield of \(\mathbb {C}\), with \(L = conj(L)\). For \(i = 1,2,3\) let \(z_i = x_i + \imath y_i \in L\) with \(z_1 \ne z_2\), and let \(r {\gt} 0\) in \(\mathbb {R}\) with \(r^2 \in L\). Define

Assume \(G \cap C \ne \emptyset \); then the following are equivalent:

• \(z\in G \cap C\).

• There is a \(\lambda \in \mathbb {R}\) with \(\lambda ^2 a+ \lambda b + c = 0\) where

  \begin{align*} a & := (x_1 - x_2)^2 + (\imath y_1 - \imath y_2)^2,\\ b & := 2(x_1 - x_2)(x_2 - x_3) - 2(\imath y_1 - \imath y_2)(\imath y_2 - \imath y_3),\\ c & := (x_2 - x_3)^2 + (\imath y_2 - \imath y_3)^2 - r^2, \end{align*}

  and \(z = \lambda z_1 + (1-\lambda )z_2\).

In this case \(z \in L(\sqrt{w})\) for an \(w \in L\).

Let \(L\) be a subfield of \(\mathbb {C}\), with \(L = conj(L)\). For \(i = 1,2,3,4\) let \(z_i = x_i + \imath y_i \in L\) with \(z_1 \ne z_2\) and \(z_3 \ne z_4\). Define

If \(G_1 \cap G_2 \ne \emptyset \) and \(G_1 \ne G_2\), it is equivalent

• \(z\in G_1 \cap G_2\).

• There are \(\lambda , \mu \in \mathbb {R}\) such that: 1. \(\lambda (x_1 - x_2)+\mu (x_4-x_3) = x_4-x_2\) 2. \(\lambda (\imath y_1 - \imath y_2)+\mu (\imath y_4-\imath y_3) = \imath y_4-\imath y_2\) 3. \(z = \lambda z_1 + (1-\lambda )z_2\)

In this case \(z \in L\).

The set of rational numbers is conjugate closed.

Let \(L\) be a subfield of \(\mathbb {C}\), with \(L = conj(L)\). For all \(z = x + \imath y \in L\) we have \(x, \imath y \in L\).

For \(\alpha \in [0,2\pi )\), if \(e^{\imath\alpha } \in M_{\infty }\), then \(e^{\imath\frac {\alpha }{2}} \in M_{\infty }\).

For \(z_1, z_2 \in M_{\infty }\) it follows that \(z_1 + z_2 \in M_{\infty }\).

For \(z \in M_{\infty }\) it follows that \(\overline{z} \in M_{\infty }\).

For \(z \in M_{\infty }\) it follows that \(z.im \in M_{\infty }\).

For \(r \in \mathbb {R}\cap M_{\infty }\) it follows that \(\imath\cdot r \in M_{\infty }\).

If \(a \in M_{\infty }\cap \mathbb {R}\), then \(a^{-1}\) is in \(M_{\infty }\).

For \(a, b \in M_{\infty }\cap \mathbb {R}\) it follows that \(a \cdot b \in M_{\infty }\).

For \(z \in M_{\infty }\) it follows that \(-z \in M_{\infty }\).

For \(l \in \mathcal{L(M_{\infty })}\) and \(z \in M_{\infty }\) it follows that \(\exists l' \in \mathcal{L(M_{\infty })}\) with \(z\in l'\) and \(l\| l'\).

If \(\)

For \(z \in M_{\infty }\) it follows that \(z.re \in M_{\infty }\).

The degree of \(K_0 = \mathbb {Q}(e^{\imath\frac {\pi }{3}},\overline{e^{\imath\frac {\pi }{3}}})\) is equal to \(2\).

Let \(K, L\) be subfield of \(\mathbb {C}\), with \(K\le L\). Then \([L:K] = 2\) is equivalent to the existence of a \(w*w \in K\) with \(w \notin K\) and \(L = K(w)\).

If two lines \(l_1\) and \(l_2\) are parallel’ and intersect, then they are equal.

For two circles \(c_1, c_2 \in \mathcal{C(M_{\infty })}\) it follows that \(c_1 \cap c_2 \in \mathcal{M}_{\infty }\).

For a line \(l \in \mathcal{L(M_{\infty })}\) and a circle \(c \in \mathcal{C(M_{\infty })}\) is \(l \cap c \in \mathcal{M}_{\infty }\).

For two lines \(l_1, l_2 \in \mathcal{L(M_{\infty })}\) is \(l_1 \cap l_2 \in \mathcal{M}_{\infty }\).

(\(\sqrt[3]{2}\) is irrational) The third root of \(2\) is an irrational number.

\(P := X^3 - 2\) is irreducible over \(\mathbb {Q}\).

For \(M\subseteq \mathbb {C}\) with \(0,1 \in M\) it holds that \(K_0 \subseteq M_{\infty }\).

The set of points of a line \(l\) is not defined by one position vector, i.e. for every \(\alpha \in l\) \(\{ \lambda x+(1-\lambda )y\mid \lambda \in \mathbb {R}\} = \{ \lambda x+\lambda y+\alpha \mid \lambda \in \mathbb {R}\} \).

For \(M\subseteq \mathbb {C}\) \(M\cup Conj(M)\) is conjugate closed.

The set \(\mathcal{M}_i\) is in \(\mathcal{M}_{\infty }\), i.e.

The set \(\mathcal{M}_i\) is monoton, i.e.

This can be generalised to

Every set \(M\) is included in the constructable points of \(M\), i.e.

The set \(\mathcal{M}\) is in \(\mathcal{M}_{\infty }\).

The set \(\mathcal{M}\) is in \(\mathcal{M}_i\), i.e.

A point \(z\) is in \(\mathcal{M}_{\infty }\) if and only if \(z\) is in \(\mathcal{M}_i\) for some \(i\).

For \(M\subseteq \mathbb {C}\) with \(0,1 \in M\) it holds:

• \(\imath\in M_{\infty }\).

• For \(z = x + \imath y \in \mathbb {C}\) the following are equivalent:

  1. \(z \in M_{\infty }\).

  2. \(x, y \in M_{\infty }\).

  3. \(x, \imath y \in M_{\infty }\).

• For \(0 \ne z = r \exp (\imath\alpha ) \in \mathbb {C}\) the following are equivalent:

  1. \(z \in M_{\infty }\).

  2. \(r,\exp (\imath\alpha ) \in M_{\infty }\).

For \(M\subseteq \mathbb {C}\) with \(0,1 \in M\), \(M_{\infty }\) is quadratic closed.

For \(M_n\) exists a chain of intermediate fields \(K_0 \le K_1 \le \ldots \le K_n\) such that \(M_i\subset K_i\) and \(K_i:= K_{i-1}(X_i)\) for a set of square roots \(X_i\) of elements of \(K_{i-1}\).

If the basis of two lines is moved by the same amount, i.e \(l_1.z_1 - l_2.z_1 = l_1.z_2- l_2.z_2\), then they are parallel.

If two lines \(l_1\) and \(l_2\) are parallel’ they are parallel.

The angle \(\pi / 3 = 60^{\circ }\) can’t be trisected using a compass and straightedge.

If \(r \in M_{\infty }\cap \mathbb {R}_{\ge 0}\), then \(\sqrt{r}\) is in \(M_{\infty }\).

For \(z \in \mathbb {C}\) there exists an \(n\ge 0\) and a chain

of subfields of \(\mathbb {C}\) such that \(z \in L_n\) and \( [L_{i+1}:L_i] = 2 \) for \(i = 0, \ldots , n-1\). This is equivalent to:
There is an intermediate field \(L\) of \(\mathbb {C}/K_0\) with \(z \in L\), so that \(L\) arises from \(K_0\) by a finite sequence of adjunctions of square roots.

A general angle can’t be trisected using a compass and straightedge.

For \(z \in \mathbb {C}\), \(z \in M_{\infty }\) is equivalent to:
There is a \(0\le n\) and a chain

of subfields of \(\mathbb {C}\) such that \(z \in L_n\) and

For \(M\subseteq \mathbb {C}\) with \(0,1 \in M\). \(M_{\infty }\) is a subfield of \(\mathbb {C}\).

The cube can’t be doubled using a compass and straightedge.