Polynomials_and_Quaternions - Polynomials and Quaternions...

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Unformatted text preview: Polynomials and Quaternions Serge Ballif, January 24, 2008 The main reference for the talk is A First Course in Noncommutative Rings by T. Y. Lam. 1 Prerequisites and Definitions 1.1 Rings Rings Definition 1. 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. Convention For the purposes of this talk, we will assume that all rings have an identity element 1 such that 1 r = r for all r R . Examples of Rings Example 2 . The following rings are familiar to all mathematicians. R the real numbers C the complex numbers Z the integers Mat n ( R ) the ring of n n real-valued matrices 1.2 Division Rings Division Rings Definition 3. A ring R is a division ring iff every nonzero element of R has a two-sided inverse element iff ( R- { } , ) is a group. Remark A field is a commutative division ring. Example 4 . R and C are division rings. Z and Mat n ( R ) are not division rings. 1.3 Quaternions The Real Quaternions Definition 5. 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 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 ....
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This note was uploaded on 04/01/2008 for the course MATH 535 taught by Professor Brownawell,woodro during the Fall '08 term at Pennsylvania State University, University Park.

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Polynomials_and_Quaternions - Polynomials and Quaternions...

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