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Polynomials_and_Quaternions_slideshow

Polynomials_and_Quaternions_slideshow - Polynomial...

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Polynomial Functions and the Quaternions Serge Ballif Prerequisites and Definitions Polynomial Functions Polynomial Functions and the Quaternions Serge Ballif [email protected] The Pennsylvania State University January 18, 2008
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Polynomial Functions and the Quaternions Serge Ballif Prerequisites and Definitions Polynomial Functions Outline Prerequisites and Definitions Rings Division Rings Quaternions Polynomial Functions and Noncommutative Rings Polynomials and Evaluation The Remainder Theorem The Factor Theorem Roots and Degree Fundamental Theorem of Algebra
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Polynomial Functions and the Quaternions Serge Ballif Prerequisites and Definitions Rings Division Rings Quaternions Polynomial Functions Rings 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. 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 .
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Polynomial Functions and the Quaternions Serge Ballif Prerequisites and Definitions Rings Division Rings Quaternions Polynomial Functions Examples of Rings Example The following rings are familiar to all mathematicians. I — the real numbers I — the complex numbers I — the integers I Mat n ( ) — the ring of n × n real-valued matrices
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Polynomial Functions and the Quaternions Serge Ballif Prerequisites and Definitions Rings Division Rings Quaternions Polynomial Functions Division Rings Definition 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 I and are division rings. I and Mat n ( ) are not division rings.
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Polynomial Functions and the Quaternions Serge Ballif Prerequisites and Definitions Rings Division Rings Quaternions Polynomial Functions The Real Quaternions Definition 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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Polynomial Functions and the Quaternions Serge Ballif Prerequisites and Definitions Rings Division Rings Quaternions Polynomial Functions Quaternion Multiplication Remark Multiplication of quaternions is not commutative. We have ij = k , but ji = - k .
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