Polynomials_and_Quaternions - Polynomials and Quaternions...

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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. — the real numbers — the complex numbers — the integers Mat n ( ) — 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 - { 0 } , × ) is a group. Remark A field is a commutative division ring. Example 4 . and are division rings. and Mat n ( ) are not division rings.
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1.3 Quaternions The Real Quaternions Definition 5. The ring of real quaternions is a 4-dimensional real vector space with basis { 1 , i, j, k } . is a division ring with multiplication structure determined by the relations i 2 = j 2 = k 2 = - 1 = ijk. An element q of is of the form q = a + bi + cj + dk, where a, b, c, d . The real part of q is a , and the purely imaginary part of q is bi + cj + dk .
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