Cyclotomic Field - Relation With Regular Polygons

Relation With Regular Polygons

Gauss made early inroads in the theory of cyclotomic fields, in connection with the geometrical problem of constructing a regular n-gon with a compass and straightedge. His surprising result that had escaped his predecessors was that a regular heptadecagon (with 17 sides) could be so constructed. More generally, if p is a prime number, then a regular p-gon can be constructed if and only if p is a Fermat prime; in other words if is a power of 2.

For n = 3 and n = 6 primitive roots of unity admit a simple expression via square root of three, namely:

ζ₃ = √3 i − 1/2, ζ₆ = √3 i + 1/2

Hence, both corresponding cyclotomic fields are identical to the quadratic field Q(√−3). In the case of ζ₄ = i = √−1 the identity of Q(ζ₄) to a quadratic field is even more obvious. This is not the case for n = 5 though, because expressing roots of unity requires square roots of quadratic integers, that means that roots belong to a second iteration of quadratic extension. The geometric problem for a general n can be reduced to the following question in Galois theory: can the nth cyclotomic field be built as a sequence of quadratic extensions?