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University of Toronto Scarborough
Sophie Chrysostomou

University of Toronto at Scarborough Department of Computer & Mathematical Sciences MAT B24S Fall 2011 Lecture 1 1 Notation A set is simply a collection of objects. We will use among other sets the following: R : the set of real numbers Z : the set of integers (positive, negative, or zero). Q : the set of all rational numbers= { p,q ∈ Z,q ▯= 0} q Zn: the set of integers from 0 to n − 1 inclusive. For exam3le Z = {0,1,2} 2 Fields Definition :A field F is a set of elements with two operations ⊕ (called addi- tion) and ⊗ (called multiplication) satisfying the following: for all a, b and c in F, the following laws hold: 1. F is closed under ⊕ and ⊗, i.e. a ⊕ b and a ⊗ b are in F whenever a and b are in F. 2. Commutative laws: a ⊕ b = b ⊕ a and a ⊗ b = b ⊗ a. 3. Associative laws: (a⊕b)⊕c = a⊕(b⊕c) and (a⊗b)⊗c = a⊗(b⊗c). 4. Distributive law: a ⊗ (b ⊕ c) = (a ⊗ b) ⊕ (a ⊗ c) 1 5. There exists an element in F called the additive identity or the zero element denoted by z such that x ⊕ z = x for all x in F. 6. For any a in F, there is an additive inverse element denoted by −a in F such that a ⊕ (−a) = z. 7. There exists an element in F called the multiplicative identity or the identity element denoted by e such that x ⊗ e = x for all x in F. 8. For any a ▯= z in F, there exists a multiplicative inverse element denoted −1 −1 by a in F such that a⊗a = e where e is the multiplicative identity defined above. Note: It is important not to assume anything about newly defined sets with addition and multiplication. It is never assumed that the identities ex- ist. They must be found and they may not be what seem to be the ”obvious” choices from the elements of the given set. Here are some examples of fields: 1. The set of the real numbers with the usual addition and multiplication. Verify this result as an exercise. 2. The set of all rational numbers with the usual addition and scalar multiplication. 3. The set Z with the addition and multiplication given in the following 3 tables: ⊕ 0 1 2 0 0 1 2 1 1 2 0 2 2 0 1 ⊗ 0 1 2 0 0 0 0 1 0 1 2 2 0 2 1 2 Note that the result of addition and scalar multiplication is the remain- der after division by 3. For example: 2+2 = 4 and 4÷3 has remainder 1. Therefore 2 ⊕ 2 = 1 (a) Note also that
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