Let R be a commutative ring with identity of prime characteristic p. If a,b R, then

Let R be a commutative ring with identity of prime characteristic p. If a,b ϵ R, then (a ± b)^pn = a^pn ± b^pn for all integers n ≥ 0 [see Theorem 1.6 and Exercise 10; note that b = -b if p = 2].

Theorem 1.6

(Binomial Theorem). Let R be a ring with identity, n a positive integer, and a,b,a₁,a₂, ... , a_s ϵ R.

(i) If ab = ba, then (a + b)ⁿ 

(ii) I∫ a_ia_j = a_ja_i for all i and j, then (a₁ + a₂...+ a_s)ⁿ =

Data from exercise 10

(a) The subset G = {1,-1,i,-i,i,-j,k,-k} of the division ring K of real quaternions forms a group under multiplication.

(b) G is isomorphic to the quaternion group.

(c) What is the difference between the ring K and the group ring R(G) (R the field of real numbers)?

Transcribed Image Text:

η Σ k = 0 k |akbn-k: