Rational Numbersand

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 0 1 2 3

1 1 2 3 0

2 2 3 0 1

3 3 0 1 2

·0 1 2 3

0 0 0 0 0

1 0 1 2 3

2 0 2 0 2

3 0 3 2 1

Finite Fields

• Is ( Z5 , + ,• ) a field? + 0 1 2 3 4

0 0 1 2 3 4

1 1 2 3 4 0

2 2 3 4 0 1

3 3 4 0 1 2

4 4 0 1 2 3

· 0 1 2 3 4

0 0 0 0 0 0

1 0 1 2 3 4

2 0 2 4 1 3

3 0 3 1 4 2

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) 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

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 NumbersVerify the field propertiesAddition Properties Multiplication PropertiesClosure ClosureIdentity IdentityInverse InverseCommutative CommutativeAssociative 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 OrderIs (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!