**Unformatted text preview: **i 2 = j 2 = k 2 =-1 = ijk. An element q of H is of the form q = a + bi + cj + dk, where a,b,c,d ∈ R . The real part of q is a , and the purely imaginary part of q is bi + cj + dk . Remark. Multiplication of quaternions is not commutative. We have ij = k , but ji =-k . i ± k 2 j b Deﬁnition. The quaternionic conjugate of q = a + bi + cj + dk is ¯ q = a-bi-cj-dk . A routine check shows that for q 6 = 0, q-1 = ¯ q a 2 + b 2 + c 2 + d 2 . Deﬁnition. For any ring R , we let R [x] denote the polynomial ring with indeterminate x and coeﬃcients from the ring R . Remark. Each polynomial f (x) ∈ R [x] is of the form f (x) = n X i =0 a i x i = a n x n + a n-1 x n-1 + ··· + a 1 x + a . 1...

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- Fall '08
- BROWNAWELL,WOODRO
- Algebra, Polynomials, Addition, cj + dk