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?
Read more about this topic: Cyclotomic Field
Famous quotes containing the words relation with, relation and/or regular:
“To criticize is to appreciate, to appropriate, to take intellectual possession, to establish in fine a relation with the criticized thing and to make it one’s own.”
—Henry James (1843–1916)
“The instincts of the ant are very unimportant, considered as the ant’s; but the moment a ray of relation is seen to extend from it to man, and the little drudge is seen to be a monitor, a little body with a mighty heart, then all its habits, even that said to be recently observed, that it never sleeps, become sublime.”
—Ralph Waldo Emerson (1803–1882)
“[I]n our country economy, letter writing is an hors d’oeuvre. It is no part of the regular routine of the day.”
—Thomas Jefferson (1743–1826)