# Lecture05 - Integral domain M5 Multiplicative Identity a1=...

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Group : Set of elements with a binary operation {G, } that adheres to these properties: A1) Closure A2) Associative A3) Identity Element (Additive), a e=e a=a A4) Inverse Element, a a'=a' a=e EXAMPLE: Permutation, On Paper A group is abelian if the following is true: A5) Commutative EXAMPLE: Integers, On Paper Cyclic Group: a 0 = e. A group is cyclic if every element of G is a power a k of a fixed element a, a is a generator of the group. A cyclic group is always abelian. (Ask to prove) Rings: {R, +, x} Use A1-A5 now add M1) Closure under multiplication M2) Associativity of multiplication M3) Distributive laws A ring is commutative if M4) commutative property of multiplication holds

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Unformatted text preview: Integral domain: M5) Multiplicative Identity a1= 1a = a M6) No zero divisors, if ab = 0 either a=0 or b=0 Fields: M7) Multiplicative Inverse Mod arithmetic A residue is in range from 0 to n-1 Two values are congruent regardless of whether they are in range or not. Basic rules of divisibility: if a | 1, a = 1 or -1 if a | b and b | a, a = b or a=-b if b!= 0 then b | 0 if b|g and b|h, then b|(gx+hy) if a ≡ b (mod n) then f(a) ≡ f(b) (mod n), where f is a poly function with integer coefficients Example: 11 7 mod 13 Residue classes: [0], [1], etc. Example of finite field of size 7 Poly arithmetic with coefficients in Z p ....
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Lecture05 - Integral domain M5 Multiplicative Identity a1=...

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