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 = abicjdk . A routine check shows that for q 6 = 0, q1 = ¯ 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 n1 x n1 + ··· + a 1 x + a . 1...
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 Fall '08
 BROWNAWELL,WOODRO
 Algebra, Polynomials, Addition, cj + dk

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