---
tag: 0SSSM1D475
kind: statement
role: theorem
name: The degree of a constructible number is a power of two
msc: [12F05]
uses: [0PN9WMRWT7, 0A7KJ68XHT, 0N5QW9K7MH, 0GK1WB9R98]
sources:
  - url: https://en.wikipedia.org/wiki/Constructible_number
    locator: Algebraic properties
---

## Statement

Let $z \in \mathbb{C}$ be a [[constructible number|0FMN2MN7TM]].
Then $z$ is algebraic over $\mathbb{Q}$, and $[\mathbb{Q}(z) : \mathbb{Q}] = 2^r$ for some integer $r \ge 0$.

## Remarks

The proof: there is a tower $\mathbb{Q} = F_0 \subseteq F_1 \subseteq \dots \subseteq F_n$ with $z \in F_n$ and $[F_j : F_{j-1}] = 2$ for each $j$. By the [[tower law|0A7KJ68XHT]], $[F_n : \mathbb{Q}] = 2^n$, and $[\mathbb{Q}(z) : \mathbb{Q}]$ divides it.
So the [[minimal polynomial|0GK1WB9R98]] of $z$ over $\mathbb{Q}$ has degree a power of two.
The converse is false: some algebraic numbers whose degree is a power of two are not constructible.
A sufficient condition: the splitting field of the minimal polynomial of $z$ has degree a power of two over $\mathbb{Q}$.

## Explanations

### L1 human

Each construction step at most doubles the degree of the field.
So a number whose minimal polynomial has degree $3$, like $\sqrt[3]{2}$, can never be constructed.
This simple test settles several problems that were open for two thousand years.