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# constructible numbers

The smallest subfield $\mathbb{E}$ of $\mathbb{R}$ over $\mathbb{Q}$ such that $\mathbb{E}$ is
Euclidean is called the *field of real constructible numbers*. First, note that $\mathbb{E}$ has the following properties:

1. $0,1\in\mathbb{E}$;

2. If $a,b\in\mathbb{E}$, then also $a\pm b$, $ab$, and $a/b\in\mathbb{E}$, the last of which is meaningful only when $b\not=0$;

3. If $r\in\mathbb{E}$ and $r>0$, then $\sqrt{r}\in\mathbb{E}$.

The field $\mathbb{E}$ can be extended in a natural manner to a subfield of $\mathbb{C}$ that is not a subfield of $\mathbb{R}$. Let $\mathbb{F}$ be a subset of $\mathbb{C}$ that has the following properties:

1. $0,1\in\mathbb{F}$;

2. If $a,b\in\mathbb{F}$, then also $a\pm b$, $ab$, and $a/b\in\mathbb{F}$, the last of which is meaningful only when $b\not=0$;

3. If $z\in\mathbb{F}\setminus\{0\}$ and $\operatorname{arg}(z)=\theta$ where $0\leq\theta<2\pi$, then $\sqrt{|z|}e^{{\frac{i\theta}{2}}}\in\mathbb{F}$.

Then $\mathbb{F}$ is the *field of constructible numbers*.

Note that $\mathbb{E}\subset\mathbb{F}$. Moreover, $\mathbb{F}\cap\mathbb{R}=\mathbb{E}$.

An element of $\mathbb{F}$ is called a *constructible number*. These numbers can be “constructed” by a process that will be described shortly.

Conversely, let us start with a subset $S$ of $\mathbb{C}$ such that $S$ contains a non-zero complex number. Call any of the binary operations in condition 2 as well as the square root unary operation in condition 3 a *ruler and compass operation*. Call a complex number *constructible from* $S$ if it can be obtained from elements of $S$ by a finite sequence of ruler and compass operations. Note that $1\in S$. If $S^{{\prime}}$ is the set of numbers constructible from $S$ using only the binary ruler and compass operations (those in condition 2), then $S^{{\prime}}$ is a subfield of $\mathbb{C}$, and is the smallest field containing $S$. Next, denote $\hat{S}$ the set of all constructible numbers from $S$. It is not hard to see that $\hat{S}$ is also a subfield of $\mathbb{C}$, but an extension of $S^{{\prime}}$. Furthermore, it is not hard to show that $\hat{S}$ is Euclidean. The general process (algorithm) of elements in $\hat{S}$ from elements in $S$ using finite sequences of ruler and compass operations is called a ruler and compass construction. These are so called because, given two points, one of which is 0, the other of which is a non-zero real number in $S$, one can use a ruler and compass to construct these elements of $\hat{S}$.

If $S=\{1\}$ (or any rational number), we see that $\hat{S}=\mathbb{F}$ is *the* field of constructible numbers.

Note that the lengths of constructible line segments on the Euclidean plane are exactly the positive elements of $\mathbb{E}$. Note also that the set $\mathbb{F}$ is in one-to-one correspondence with the set of constructible points on the Euclidean plane. These facts provide a connection between abstract algebra and compass and straightedge constructions.

## Mathematics Subject Classification

12D15*no label found*

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