Ruler-and-compass geometry, roots of unity, and the deep algebra beneath classical problems
Why can a ruler and compass build a regular 17-gon but not trisect an angle? Galois theory answers questions the ancient Greeks could not.
Draw lines and circles, then press Find Intersections to make new points. Try the challenges: build sqrt(2) and a regular pentagon, then see where trisecting 60 degrees gets stuck.
Construct geometric objects from two initial points. Discover which field extensions emerge.
Classical ruler-and-compass constructions are one of the oldest problems in mathematics. Galois theory reveals the deep algebraic structure beneath: a length is constructible if and only if it lies in a tower of quadratic extensions of Q -- so its minimal polynomial must have degree a power of 2. This explains why trisecting an angle or doubling a cube is impossible, and why Gauss could construct the regular 17-gon.
Use classical tools -- straight lines and circles -- to construct new points from intersections. Each intersection gives coordinates in a field extension of the previous one, with degree at most 2 (since circles are degree-2 curves). Try the challenges to see what is and is not constructible.
Key insight: Each ruler-compass step extends the coordinate field by at most degree 2 (intersecting a circle with a line or another circle). So every constructible number lives in an extension of degree 2^k over Q. This is why you cannot trisect 60 degrees: cos(20 degrees) satisfies a cubic with no rational roots, and 3 does not divide any power of 2.
Pick n to see the nth roots of unity on the circle, then choose an automorphism to watch the roots permute.
Explore nth roots of unity and their Galois automorphisms on the unit circle.
Cyclotomic fields -- the fields generated by roots of unity -- provide the bridge. Their Galois groups are the unit groups (Z/nZ)*, which are always abelian and often have power-of-2 structure.
The nth roots of unity are equally spaced on the unit circle. Their minimal polynomial is the cyclotomic polynomial, and the Galois group Gal(Q(zeta_n)/Q) is isomorphic to (Z/nZ)*, the multiplicative group of integers mod n. Select an automorphism to watch the roots permute.
Key insight: The automorphism sigma_k sends zeta_n to zeta_n^k. Since (Z/nZ)* is always abelian, every cyclotomic extension is an abelian Galois extension. This is why cyclotomic fields are so well-behaved -- the Kronecker-Weber theorem says that every abelian extension of Q is contained in some cyclotomic field.
Click any polygon to see its analysis. Solid outlines are constructible and dashed ones are not; find Gauss's 17-gon.
Regular n-gons for n = 3 to 20. Solid purple = constructible, dashed gray = not constructible.
A regular n-gon is constructible by ruler and compass if and only if n = 2^k * p_1 * p_2 * ... * p_m, where the p_i are distinct Fermat primes (primes of the form 2^(2^j) + 1). The known Fermat primes are 3, 5, 17, 257, and 65537. Click any polygon to see the analysis.
Key insight: The constructibility criterion comes from the Galois group of Q(zeta_n)/Q. A regular n-gon is constructible iff phi(n) is a power of 2, which happens exactly when n has the Fermat-prime factorization above. Gauss's construction of the 17-gon (1796) was a triumph -- he showed that phi(17) = 16 = 2^4, so the construction requires exactly 4 square-root steps.
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