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Rational Numbers and Fields Integers – Well ordered integral domain •Can we solve any linear equation over the integers? Example: x + 5 = 7 3x + 5 = 11 •What property do the integers lack that we need to be able to solve the equation on the right? Field • A commutative ring F with unity where every nonzero element of F has a multiplicative inverse in F. • F must also have more than one element. Why? Discussion • Give at least two examples of fields. Finite Fields • Must a field be an infinite set? Let’s explore. • Is ( Z4 , + ,• ) a field? + 0 1 2 3 · 0 1 2 3 0 0 1 2 3 0 0 0 0 0 1 1 2 3 0 1 0 1 2 3 2 2 3 0 1 2 0 2 0 2 3 3 0 1 2 3 0 3 2 1 Finite Fields • Is ( Z5 , + ,• ) a field? + 0 1 2 3 4 · 0 1 2 3 4 0 0 1 2 3 4 0 0 0 0 0 0 1 1 2 3 4 0 1 0 1 2 3 4 2 2 3 4 0 1 2 0 2 4 1 3 3 3 4 0 1 2 3 0 3 1 4 2 4 4 0 1 2 3 4 0 4 3 2 1 Finite Fields • (Z p , + , • ) where p is prime is a field. Proof: Verify it is a commutative ring with unity (you can do this). Verify the existence of an inverse. Division Algorithm for Integers • Let a, b Z with b 0, then there exist unique q, r Z such that a = b•q + r where 0 < r < | b | • We name these integers the Dividend a, Divisor b, Quotient q, Remainder r • Example: 25/7 can be expressed 25 = 7 • 3 + 4 where 0 < 4 < 7 Euclidean Algorithm • Greatest Common Divisor (g.c.d) can be found by repeated application of the Division Algorithm. • Example: gcd(630,66) 630 = 66•9 + 36 66 = 36•1 + 30 36 = 30•1 + 6 30 = 6•5 + 0 Generalization: gcd(a1, a2 ) a1 = a2 • q1 + a3 a2 = a3 • q2 + a4 a3 = a4 • q3 + a5 an-2 = an-1 • qn-2 + an an-1 = an • qn-1 Finite Fields • The Euclidean Algorithm provides the existence of the inverse in Zp Proof (completed): We needed ax + p(-q) = 1. Since p is prime then gcd(a, p) = 1. So by the Euclidean Algorithm an-2 = an-1 • qn-2 + 1 or an-2 - an-1 • qn-2 = 1 We can back substitute for the a values to get the desired equation. QED Field and Integral Domain • Is a field F always an integral domain? • Verify this by letting r,s F such that r • s = 0 and suppose r 0. What do we have to show? Rational Numbers – An Extension of the Integers • Let S = {(a , b) | a , b Z ,b 0 } • Think of (a , b) as familiar a / b, but symbol a / b has no meaning until there is a field containing a and b. • Want a / b = a•n / b•n for any n Z, n0. So need (a ,b) (an ,bn) Rational Numbers – An Extension of the Integers • Define equivalence relation (a ,b) (c,d) only if ad = bc. • Verify this is an equivalence relation. • Consider Equivalence Classes [a, b] = {(x ,y) | (x ,y) S and ay = bx} • Provide an example of an equivalence class • Let our new field F = { [a, b ] | (a ,b) S} Binary Operations on set F = { [a ,b] | (a , b) S } • Define so they parallel + and • of rational numbers • Addition: [a ,b] + [c , d] =[ad+bc,bd] • Multiplication: [a, b] • [c ,d] = [ac,bd] • Closure: For all x , y Set, x+y Set and x • y Set. Well Defined Operation: If X = X1 and Y = Y1 then X + Y = X1+ Y1 If X = X1 and Y = Y1 then X • Y = X1 • Y1 (F,+,•) field of Rational Numbers Verify the field properties Addition Properties Multiplication Properties Closure Closure Identity Identity Inverse Inverse Commutative Commutative Associative Associative Distributive Property Quotient Field • What is the additive identity? • What is the additive inverse? Quotient Field • What is the Multiplicative Identity? • What is the Multiplicative Inverse? Question • In extending D to F, why is it necessary that D be an integral domain, and not just a commutative ring with unity? Rational Order Is (Q,+,•) an ordered integral domain? Recall the definition of ordered. Ordered Integral Domain: Contains a subset D+ with the following properties. 1. If a, b D+ ,then a + b D+ (closure) 2. If a , b D+ , then a • b D+ 3. For each a Integral Domain D exactly one of these holds a = 0, a D+ , -a D+ (Trichotomy) • Thank you!