Compass, straightedge and powers of two
Which regular polygons can be built with compass and straightedge, what the degree of an algebraic number is, and whether a segment of a given length can be constructed.
How to solve it
Compass and straightedge can add, subtract, multiply, divide and take square roots — nothing more. So a constructible number always comes from a chain of quadratic extensions, and the degree of its minimal polynomial is a power of two. Hence the cube cannot be doubled: $\sqrt[3]{2}$ has degree $3$.
Step by step
- The degree of $\alpha$: find the polynomial with rational coefficients of the smallest degree that has it as a root (the minimal one). For $\sqrt[n]{a}$ it is usually $x^n - a$.
- Constructibility: if the degree is not a power of two, it cannot be constructed.
- A regular $n$-gon: factor $n$. It can be built only if $n$ is a power of two times distinct Fermat primes $3, 5, 17, 257, 65537$.
- A repeated Fermat prime ($9 = 3^2$, $25 = 5^2$) or another odd prime ($7$, $11$, $13$) makes the construction impossible.
Common mistakes
- Calling an $n$-gon constructible when all its prime factors are Fermat primes, forgetting repeats: a $9$-gon is not constructible.
- Thinking any number with roots is constructible: $\sqrt[3]{5}$ is a root, but a cube root of degree $3$.
- Using a polynomial that is not minimal: $\sqrt{2} + \sqrt{3}$ has degree $4$, although the roots look quadratic.
- Reversing the statement: a power-of-two degree does not by itself guarantee constructibility.