1. Rational Numbers and Fields

2. Integers – Well ordered integral domain
Can we solve any linear equation over the integers?
Example: x + 5 = x + 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. Finite Fields Must a field be an infinite set? Let’s explore.
Is ( Z4 , + ,• ) a field? + 1 2 3 1 2 3

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

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 • 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• a1 = a2 • q1 + a3
66 = 36• a2 = a3 • q2 + a4
36 = 30• 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 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!