Download Lecture 5

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?
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Finite Fields
• Is ( Z5 , + ,• ) a field?
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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, n0.
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!