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)
“A theory of the middle class: that it is not to be determined by its financial situation but rather by its relation to government. That is, one could shade down from an actual ruling or governing class to a class hopelessly out of relation to government, thinking of gov’t as beyond its control, of itself as wholly controlled by gov’t. Somewhere in between and in gradations is the group that has the sense that gov’t exists for it, and shapes its consciousness accordingly.”
—Lionel Trilling (1905–1975)
“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)