Ruler-and-Compass Constructions

For two thousand years, geometers chased three problems with nothing but an unmarked straightedge and a compass: double the cube (build a cube of twice the volume), trisect an arbitrary angle, and square the circle (build a square of the same area as a given circle). Generations tried. All three are impossible — and the proof is not geometric at all. It is a single, startling idea from field theory:

A length you can build with ruler and compass, starting from a unit segment, always lives in a field reached from \mathbb{Q} by a tower of degree-2 extensions. And so — by the tower law — its degree over \mathbb{Q} must be a power of two. Any number whose degree is not a power of two is not constructible, full stop. The three ancient puzzles ask for numbers of degree 3 or worse, and 3 is not a power of 2. Two millennia of frustration, dissolved in one line of arithmetic.

What a construction can do, algebraically

Set up coordinates so the given unit length is the segment from (0,0) to (1,0). A ruler-and-compass construction proceeds one step at a time, and each step is one of exactly three moves: draw a line through two known points, draw a circle centred at a known point through another, or mark an intersection of two such lines/circles. New points are born only at those intersections.

Now watch the algebra. A line through known points has a linear equation with coordinates in the current field K. A circle has a quadratic equation with coefficients in K. Intersecting line-with-line stays in K; intersecting line-with-circle or circle-with-circle means solving a quadratic, so the new coordinates land in K or in K(\sqrt{d}) for some d \in K. Every new point therefore lives in an extension of degree 1 or 2 over the last one.

Chaining the steps builds a tower \mathbb{Q} = K_0 \subseteq K_1 \subseteq \cdots \subseteq K_m with each [K_{i+1} : K_i] \in \{1, 2\}. By the tower law the total degree multiplies:

[K_m : \mathbb{Q}] = \prod_{i} [K_{i+1} : K_i] = 2^k

for some k. A constructible number \alpha sits inside such a K_m, so [\mathbb{Q}(\alpha) : \mathbb{Q}] divides 2^k and is itself a power of two. That is the entire toolkit.

The three impossibilities

A real number \alpha is constructible if and only if it lies in a field obtained from \mathbb{Q} by a finite tower of degree-2 extensions.
Consequently, if \alpha is constructible then [\mathbb{Q}(\alpha) : \mathbb{Q}] is a power of 2.
Contrapositive — the workhorse: if [\mathbb{Q}(\alpha) : \mathbb{Q}] is not a power of 2, then \alpha cannot be constructed.

Doubling the cube. Two copies of a unit cube have volume 2, so the new side length is \sqrt[3]{2}. Its minimal polynomial over \mathbb{Q} is x^3 - 2 — irreducible by Eisenstein at p = 2 — so [\mathbb{Q}(\sqrt[3]{2}) : \mathbb{Q}] = 3. Three is not a power of two, so \sqrt[3]{2} is not constructible. The Delian problem is impossible.

Trisecting a general angle. Try 60^\circ, which itself is constructible. Trisecting it needs \cos 20^\circ. The triple-angle identity \cos 3\theta = 4\cos^3\theta - 3\cos\theta with 3\theta = 60^\circ gives, after setting x = 2\cos 20^\circ,

8x^3 - 6x - 1 = 0,

which is irreducible over \mathbb{Q} (no rational root). So \cos 20^\circ has degree 3 — again not a power of two, so 60^\circ cannot be trisected with these tools, and hence no method trisects every angle.

Squaring the circle. A circle of radius 1 has area \pi, so the square's side is \sqrt{\pi}. But Lindemann proved in 1882 that \pi is transcendental — it is not a root of any polynomial with rational coefficients. So \sqrt{\pi} is not algebraic at all; its "degree" is infinite, certainly not a power of two. Squaring the circle is impossible — the deepest of the three, because it needed a brand-new theorem about the nature of \pi.

Gauss and the buildable polygons

The theory does not only forbid — it also grants. As a teenager, Gauss discovered that the regular 17-gon is constructible, the first new constructible polygon in two thousand years, and he was so proud he asked for one on his tombstone. The full criterion he opened (finished by Wantzel) is beautiful:

A regular n-gon is constructible if and only if n = 2^k \, p_1 p_2 \cdots p_r, where the p_i are distinct Fermat primes — primes of the form 2^{2^m} + 1.

The known Fermat primes are 3, 5, 17, 257, 65537. So the 3-, 5-, 15-, 16-, and 17-gons are all constructible — but the regular 7-gon is not (7 is not a Fermat prime), and neither is the regular 9-gon (9 = 3^2 repeats the Fermat prime 3). The same power-of-two engine, running in reverse.

It is tempting to flip the theorem into "degree 2^k \Rightarrow constructible." That is false. There exist numbers of degree 4 that are not constructible. The catch is subtle: constructibility demands a tower where every step is degree 2, and for that the whole splitting field must have degree a power of two (equivalently, a Galois group of 2-power order). A number can have degree 4 while its splitting field has degree, say, 24 — no all-quadratic tower reaches it. So "degree is a power of 2" is a necessary test that quickly kills the classical problems, but it is not the complete sufficient condition.

A second trap: "ruler" here means an unmarked straightedge. If you allow a marked ruler and the sliding trick called neusis, you can trisect any angle and double the cube — Archimedes did. The impossibility is a statement about a specific, deliberately minimal toolkit, not about geometry itself.

Even after Wantzel (1837) and Lindemann (1882) closed the book, amateur "angle trisectors" kept mailing proofs to mathematics departments — so many that Augustus De Morgan wrote a whole book, A Budget of Paradoxes, cataloguing them. The phrase "squaring the circle" survives in everyday English as a byword for attempting the impossible. The lesson the field-theory proof teaches is worth the two thousand years: knowing what cannot be done is as much a triumph as a construction, and the right language — degrees of extensions — can make an ancient wall of difficulty simply vanish.