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:
“[Man’s] life consists in a relation with all things: stone, earth, trees, flowers, water, insects, fishes, birds, creatures, sun, rainbow, children, women, other men. But his greatest and final relation is with the sun.”
—D.H. (David Herbert)
“Whoever has a keen eye for profits, is blind in relation to his craft.”
—Sophocles (497–406/5 B.C.)
“While you’re playing cards with a regular guy or having a bite to eat with him, he seems a peaceable, good-humoured and not entirely dense person. But just begin a conversation with him about something inedible, politics or science, for instance, and he ends up in a deadend or starts in on such an obtuse and base philosophy that you can only wave your hand and leave.”
—Anton Pavlovich Chekhov (1860–1904)