Answer to Question #16873 in Algebra for sanches

Question

Let a be an element in a ring such that ma = 0 = a^(2^r) , where m ≥ 1 and r ≥ 0 are given integers. Show that (1 + a)^(m^r) = 1.

Answer 1

The proof is by induction on r.The case r = 0 being clear, we assume r > 0. Since ma =0, the binomial theorem gives (1 + a)^m = 1+a²bwhere b is a polynomial in a with integer coefficients. Sincem(a²b) = 0 and (a²b)^2^r−1= a^2^r*b^2^r−1 = 0, theinductive hypothesis (applied to the element a²b)implies that 1 = (1+a²b)^m^r−1= [(1 + a)^m]^m^r−1 = (1+a)^m^ras desired.