May 19th, 2015
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modulus, multiplicative inverse, fields, and Rings

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Example.(Addition and multiplication mod 5)Construct addition and multiplication tables for.Notice in the multiplication table thatThis means that,, and. In particular, to divide by 2 you multiply by 3: in, the elements 2 and 3 arereciprocals.Commutative rings with identity come up in discussingdeterminants, but the algebraic system of greatest importance in linear algebra is thefield.Definition.Let R be a ring with identity, and let. Themultiplicative inverseof x is an elementwhich satisifiesDefinition.AfieldF is a commutative ring with identity in whichand every nonzero element has a multiplicative inverse.By convention, you don't write "" for "" unless the ring happens to be,, or.If an element x has a multiplicative inverse,you can divide byxby multiplying by. Thus, in a field, you can divide by any nonzero element. (You'll learn in abstract algebra why it doesn't make sense to divide by 0.)Example.The rationals, the reals, and the complex numbersare fields. Many of the examples will use these number systems.The ring of integersis not a field. For example, 2 is a nonzero integer, but it does not have a multiplicative inverse {\it which is an integer}. (is not an integer --- it's a rational number.),, andare all infinite fields --- that is, they all have infinitely many elements. For applications, it's important to considerfinitefields as well

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