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**Unformatted text preview: **Field Axioms
Definition 1. A binary operation, , on a set A, is a function :AAA Note: has to be defined for every pair a, b A, and give an element of A input is an ordered pair the value on an ordered pair (a, b) is written a b Example. On Z, +, -, and , but not . Definition 2. If is a binary operation on A, then is commutative if ab=ba is associative if a (b c) = (a b) c has an identity element e if ae=ea=a for all a A if has an identity e and a A, we say that a has an inverse if there exists b A s.t. ab=ba=e for all a, b, c A for all a, b A Definition 3. A set F with two binary operations + and is a field if (1) + and are commutative (2) + and are associative (3) + and have identities, which we will denote by 0 and 1 respectively, and 0 = 1 (4) every element a F has an inverse for +, written -a, and every element a F - {0} has an inverse for , written a-1 (5) Distributive law holds: a (b + c) = a b + a c ...

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