Rational Numbers and Fields

1. Rational Numbersand Fields

2. 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?

3. 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?

4. Discussion • Give at least two examples of fields.

5. · + 0 0 1 1 2 2 3 3 0 0 0 0 1 0 0 2 3 0 1 1 0 1 1 2 2 3 3 0 2 2 0 2 2 3 0 0 2 1 3 3 3 0 0 3 2 1 1 2 Finite Fields • Must a field be an infinite set? Let’s explore. • Is ( Z4 , + ,• ) a field?

6. + · 0 0 1 1 2 2 3 3 4 4 0 0 0 0 0 1 0 2 3 0 4 0 1 1 0 1 1 2 2 3 3 4 0 4 2 2 0 2 3 2 4 4 1 0 1 3 3 3 0 3 3 4 0 1 4 1 2 2 4 4 0 4 4 0 1 3 2 2 1 3 Finite Fields • Is ( Z5 , + ,• ) a field?

7. 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.

8. 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

9. Euclidean Algorithm • Greatest Common Divisor (g.c.d) can be found by repeated application of the Division Algorithm. • Example: gcd(630,66) Generalization: gcd(a1, a2 ) 630 = 66•9 + 36 a1 = a2• q1 + a3 66 = 36•1 + 30 a2 = a3• q2 + a4 36 = 30•1 + 6 a3 = a4• q3 + a5 30 = 6•5 + 0 an-2 = an-1• qn-2 + an an-1 = an• qn-1

10. 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

11. 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?

12. 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, n0. So need (a ,b)  (an ,bn)

13. 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}

15. 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]

16. 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

17. (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

18. Quotient Field • What is the additive identity? • What is the additive inverse?

19. Quotient Field • What is the Multiplicative Identity? • What is the Multiplicative Inverse?

20. Question • In extending D to F, why is it necessary that D be an integral domain, and not just a commutative ring with unity?

21. Rational Order Is (Q,+,•) an ordered integral domain? Recall the definition of ordered. Ordered Integral Domain: Contains a subset D+ with the following properties. If a, b  D+ ,then a + b  D+ (closure) If a , b  D+ , then a • b  D+ For each a  Integral Domain D exactly one of these holds a = 0, a  D+ , -a  D+ (Trichotomy)

22. Thank you!