Polynomials_and_Quaternions_handout

Polynomials_and_Quaternions_handout - i 2 = j 2 = k 2 =-1 =...

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Polynomials and Quaternions The main reference for the talk is A First Course in Noncommutative Rings by T. Y. Lam. Definition. A ring R is a set together with two binary operations “+” and “ × ” such that the following three properties hold. 1. ( R, +) form an additive group. 2. Multiplication is associative (i.e. ( ab ) c = a ( bc ) for all a,b,c R ). 3. The left and right distributive laws hold: a ( b + c ) = ab + ac and ( a + b ) c = ac + bc. We will assume that all rings have an identity element 1 such that 1 r = r for all r R . A ring R is a division ring iff it satisfies 4. every nonzero element of R has a two-sided inverse element (i.e. ( R - { 0 } , × ) is a group). Example. A field is a commutative division ring. R and C are division rings. Z and Mat n ( R ) are not division rings. Definition. The ring of real quaternions H is a 4-dimensional real vector space with basis { 1 ,i,j,k } . H is a division ring with multiplication structure determined by the relations
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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 Denition. 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 . Denition. For any ring R , we let R [x] denote the polynomial ring with indeterminate x and coecients 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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