A finite division ring Dis a field. Here is an outline of the proof (in which E* denotes the multiplicative group of nonzero elements of a division ring E).

(a) The center K of Dis a field and D is a vector space over K, whence IDI = qⁿ where q = IKI ≥ 2.

(b) If O ≠ a ϵ D, then N(a) = {d ϵ D | da = ad} is a subdivision ring of D containing K. Furthermore, IN(a)I = q^r where r|n.

(c) If ≠ a ϵ D - K, then N(a)* is the centralizer of a in the group D* and [D* : N(a)*] = (qⁿ - 1)/(q^r - 1) for some r such that 1 ≤ r ≤ n and r | n 

(d)

where the last sum taken over a T finite number of integers r such that 1≤ r

(e) For each primitive nth root of unity ζ ϵ C, |q - ζ| > q - 1, where

Consequently, |g_n(q)I > q - 1, where g_n is the nth cyclotomic polynomial over Q.

(f) The equation in (d) is impossible unless n = 1, whence K = D.

Transcribed Image Text:

- q₁ − 1 = q = 1+ 2q - 1)/(q − 1), - T