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8/4/2019 Chapter v - Algebraic Structures II
1/31
Rings and fiels
Algebraic structures - Part II
NGUYEN CANH Nam1
1Faculty of Applied MathematicsDepartment of Applied Mathematics and Informatics
Hanoi University of Technologies
namnc@mail.hut.edu.vn
HUT - 2010
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Rings and fiels
Agenda
1 Rings and fiels
Rings
Fields
Ring of integers
Euclidean Algorithm
Presentation of integers
NGUYEN CANH Nam Mathematics I - Chapter 5
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Rings and fiels
Rings
Fields
Ring of integers
Euclidean Algorithm
Presentation of integers
Agenda
1 Rings and fiels
Rings
Fields
Ring of integers
Euclidean Algorithm
Presentation of integers
NGUYEN CANH Nam Mathematics I - Chapter 5
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Rings and fiels
Rings
Fields
Ring of integers
Euclidean Algorithm
Presentation of integers
RingsDefinition
Definition
Let V be a set with two binary operators usually as addition andmultiplication (+ and ). Then V is a ring if
i) (V, +) is an Abelian group with the identity 0.
ii) (closure for multiplication) If x, y V then x y V.
iii) (associative law) For x, y, z V, (x y) z = x (y z),
iv) (distributive laws) For x, y, z V,x (y + z) = x y + x z, (x + y) z = x z + y z.
If V has the property : x y = y x for x, y V then V is called
commutative.
If V has the identity for the multiplication, V is called a ring with identity.NGUYEN CANH Nam Mathematics I - Chapter 5
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Rings and fiels
Rings
Fields
Ring of integers
Euclidean Algorithm
Presentation of integers
RingsExamples
Example
a) (ZZ, +, ), (OQ, +, ), (IR, +, ) are commutative rings with identity.
b) Let E be the set of even integers with usual addition andmultiplication. We have known that (E, +) is an Abelian group.Since the associativity, distributivity and commutativity hold forall integers and, therefore, are true whenever a, b, c are even.Consequently, E is a commutative ring. E does not have an
identity, however, because no even integer e has the propertythat ae = a= eafor every even integer.
c) The set of odd integers with the usual addition and multiplicationis not a ring. Because, as we have known, the set of oddintegers is not closed under addition.
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Rings and fiels
Rings
Fields
Ring of integers
Euclidean Algorithm
Presentation of integers
RingsExamples
Example
e) Let X be a non-empty set. Let R be the set of mappingsfrom X to IR. We introduce an addition and multiplication
into R as follows. Let f : X IR and g : IR. Define thesum f + g and the product f g as mapping from X to IR by,
(f + g)(x) = f(x) + g(x) (x X),
(f g)(x) = f(x)g(x) (x X).
We should note that on the right-hand side of the first
equation the plus sign + refers to addition in IR and on theright-hand side of the second equation the product is in IR.
R is a commutative ring.NGUYEN CANH Nam Mathematics I - Chapter 5
Ri
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Rings and fiels
Rings
Fields
Ring of integers
Euclidean Algorithm
Presentation of integers
Agenda
1 Rings and fiels
Rings
Fields
Ring of integers
Euclidean Algorithm
Presentation of integers
NGUYEN CANH Nam Mathematics I - Chapter 5
Rings
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Rings and fiels
Rings
Fields
Ring of integers
Euclidean Algorithm
Presentation of integers
FieldsDefinition
Definition
Let F be a ring. We say that F is a field if
i) F is a commutative ring with identity 1.
ii) For x= 0, y= 0, we have x y= 0
iii) For every x= 0, there exists the inverse element x1 suchthat x x1 = 1.
Remark
If F is a field, then F = F\ {0} is group under the multiplication
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Rings and fiels
Rings
Fields
Ring of integers
Euclidean Algorithm
Presentation of integers
FieldsExamples
Example
a) (IR,+, ) is a field,
b) (OQ,+, ) is a field.
c) Z3 = {0, 1, 2}, where0 = {3n | n ZZ}, 1 = {3n+1 | n ZZ}, 2 = {3n+2 | n ZZ}.
Defind addition and multiplication operations as followsi + j = i + j, i j = i j. Then Z3 is a field.
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Rings and fiels
Rings
Fields
Ring of integers
Euclidean Algorithm
Presentation of integers
CharacteristicDefinition
Let F be a field, x is an element of F the sum
x + x + + x (k terms )
is denoted by kx.
Definition
Let F be a field, e be the identity element of F. If p is the
smallest natural number such that pe = 0 then p is called thecharacteristic of the field F. If ke= 0 for every natural number kthen the characteristic is zero.
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Rings and fiels
Rings
Fields
Ring of integers
Euclidean Algorithm
Presentation of integers
CharacteristicExamples
Example
a ) Z3 = {0, 1, 2}, since 1 + 1 + 1 = 3.1 = 0 . Characteristicof Z3 is 3.
b) Field IR of real numbers has the characteristic zero.
c) Field OQ of rational numbers has the characteristic zero.
Theorem
Assume that p is the characteristic of a field. If p= 0 then p is aprime number.
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Rings and fiels
Rings
Fields
Ring of integers
Euclidean Algorithm
Presentation of integers
Agenda
1 Rings and fiels
Rings
Fields
Ring of integers
Euclidean Algorithm
Presentation of integers
NGUYEN CANH Nam Mathematics I - Chapter 5
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Rings and fiels
g
Fields
Ring of integers
Euclidean Algorithm
Presentation of integers
DivisorsDefinition
Definition
Let m and n be integers with b being nonzero. We say that m
divides n and write m | n if there exists an integer k such that
n = km. We also say in this case m is a divisor of n and n is amultiple of m.
Notice that 0 is divisible by any integer b, b= 0, since 0 = b0and that any integer a= 0 has so-called trivial divisors 1,a.
Example
a) 3 is a divisor of 6, and 6 is a multiple of 3. We denote 3 | 6.
b) 7 has the trivial divisors 1,7. There are no otherdivisors.
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Rings and fiels
g
Fields
Ring of integers
Euclidean Algorithm
Presentation of integers
Binary operators
Definition
A strictly positive integer p= 1 is said to be a prime number or,simply, a prime if p has only trivial divisors.
Theorem
If a, d are natural numbers, d is a nonzero then there existunique unique integers q and r such that
a= qd + r, 0 r < d.
The number q is called the quotient and r is called the
remainder. If r = 0 then d is a divisor of aand a is a multiple ofd.
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Rings and fiels
Fields
Ring of integers
Euclidean Algorithm
Presentation of integers
Congruence
Definition
Let n be a given non zero integer. Two integers aand b are said to becongruent modulo n if a b is a multiple of n. We write
a b mod m.
Otherwise we say that a is not congruent to b modulo n if n does notdivide a b.
Remark
Two integers a, b are congruent modulo m if and only if they have the
same remainder after dividing by the modulo m.
Example
a) 3 8 mod 5,
b) 3 17 mod 10.NGUYEN CANH Nam Mathematics I - Chapter 5
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Rings and fiels
Fields
Ring of integers
Euclidean Algorithm
Presentation of integers
Congruence
Proposition
The congruence modulo m relation is an equivalence relation.
Proposition
If a b mod m and c d mod m then
i) (a+ c) (b+ d) mod m,
ii) (a c) (b d) mod m,
iii) ac bd mod m.
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Rings and fiels
Fields
Ring of integers
Euclidean Algorithm
Presentation of integers
Congruence
Definition
The set of integers congruent to an integer i modulo m is called
the congruence class of i modulo m. This class is denote by i.
The set of congruence class modulo m is denoted by ZZm orZZ/mZZ. Thus ZZm = {0, 1, 2, , m 1}.Now we define the addition and multiplication on ZZm as follows
a+ b = a+ b, a.b = ab
Proposition
ZZm with the above addition and multiplication becomes a
commutative ring.
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Fi ld
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Rings and fiels
Fields
Ring of integers
Euclidean Algorithm
Presentation of integers
Agenda
1 Rings and fiels
Rings
Fields
Ring of integers
Euclidean Algorithm
Presentation of integers
NGUYEN CANH Nam Mathematics I - Chapter 5
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Fields
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Rings and fiels
Fields
Ring of integers
Euclidean Algorithm
Presentation of integers
gcdDefinitions
Definition
Let aand b be integers, not both 0. The greatest common
divisor (gcd) of aand b is the largest integer d that divides both
aand b. In other words, d is the gcd of aand b provided that
i) d | aand d | b,
ii) if c | aand c | b then c d.
Denote the greatest common divisor of aand b by GCD(a, b).
Definition
If GCD(a, b) = 1 then a, b are said to be coprime or relativelyprime.
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Fields
Ring of integers
Euclidean Algorithm
Presentation of integers
gcdExamples
Example
a) GCD(6, 9) = 3,
b) GCD(180, 315) = 45,c) GCD(315, 143) = 1. So 315 and 143 are copime.
Proposition
Let a and b be integers, not both 0, and GCD(a, b) = d. Thereexists integers m, n (not necessarily unique) such thatam+ bn = d. Furthermore, d is the smallest positive integerthat can be written in the form am+ bn.
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Rings and fiels
Fields
Ring of integers
Euclidean Algorithm
Presentation of integers
lcdDefinition
Definition
For two natural numbers a, b the lowest common denominator(lcd) is the least common multiple of aand b.
Example
a) LCD(6, 9) = 18,b) LCD(180, 315) = 1260.
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Rings and fiels
Fields
Ring of integers
Euclidean Algorithm
Presentation of integers
gcd and lcdProperties
Proposition
For natural numbers a, b we have
ab = GCD(a, b).LCD(a, b).
Proposition
Suppose that natural numbers a, b, q, r satisfy the formula
a= bq+ r.
Then
GCD(a, b) = GCD(b, r).
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Rings and fiels
Fields
Ring of integers
Euclidean Algorithm
Presentation of integers
Euclidean Algorithm.
Introduction
For any integer b, we know that b and b have the samedivisors. Consequently, the common divisors of aand b are the
same as the common divisors of aand b. Therefore, the
greatest common divisors must be the same, that is,GCD(a, b) = GCD(a,b). Using similar arguments, we seethat
GCD(a, b) = GCD(a,b) = GCD(a, b) = GCD(a,b).
So a method for finding the gcd of two positive integers can be
also used to find the gcd of any two integers.
Moreover, from the above proposition one can obtain a method
of finding GCD of aand b called Euclidean algorithm.
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Rings and fiels Ring of integers
Euclidean Algorithm
Presentation of integers
Euclidean Algorithm.
Algorithm
Let aand b be positive integers with a b. If b | a thenGCD(a, b) = b. Otherwise apply the division algorithmrepeatedly as follows:
a= q0b+ r0, 0 < r0 < b
b = q1r0 + r1, 0 r1 < r0r0 = q2r1 + r2, 0 r2 < r1
r1 = q3r2 + r3, 0 r3 < r2
...
This process ends when a remainder of 0 is obtained. This
must occur after a finite number of steps; that is, for someinteger t :
rt2 = qtrt1 + rt, 0 < rt < rt1
rt1 = qt+1rt + 0
Then rt, the last nonzero remainder, is the greatest common
divisor of aand b.NGUYEN CANH Nam Mathematics I - Chapter 5
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Rings and fiels Ring of integers
Euclidean Algorithm
Presentation of integers
Euclidean Algorithm.
Example
Example
Find gcd of 1071 and 1029.
Solution.
1071 = 1 1029 + 42,
1029 = 24 42 + 21,
42 = 2 21.
Hence the greatest common divisor of 1071 and 1029 is 21.
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Rings and fiels Ring of integers
Euclidean Algorithm
Presentation of integers
Agenda
1 Rings and fiels
RingsFields
Ring of integers
Euclidean Algorithm
Presentation of integers
NGUYEN CANH Nam Mathematics I - Chapter 5
RingsFields
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Rings and fiels Ring of integers
Euclidean Algorithm
Presentation of integers
Base 10
Introduction
We usually write integers as form 10-adic. For example,
2139 = 2.103 + 1.102 + 3.101 + 9.100.
Given a positive integer b. For a natural number n we have the
expression
n = akbk + ak1b
k1 + + a1.b+ a0, 0 aj < b, ak = 0. ()
Then the presentation () is said to be the expansion of n bybase b, denoted by n = (akak1...a1a0)b.If b = 2 the presentation () is called the binary expansion of n.
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Euclidean Algorithm
Presentation of integers
Bases
Examples
Example
a) The binary expansion of 35 is
35 = 1 25 + 0 24 + 0 23 + 0 22 + 1 21 + 120
35 = (100011)2.
b) The expansion of 135 by base 4 is
135 = 2 43 + 0 42 + 1 41 + 3 40
135 = (2013)4.
NGUYEN CANH Nam Mathematics I - Chapter 5
Ri d fi l
RingsFields
Ri f i
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Rings and fiels Ring of integers
Euclidean Algorithm
Presentation of integers
Algorithm for expansion of n by base b.
We now present an algorithm for finding the expansion of a
positive integer n by base b. The algorithm is proceeded
repeatedly as follows:
n = bq0 + r0,
q0 = bq1 + r1,
q1 = bq2 + r2
...
This process ends when a quotient of 0 is obtained. This mustoccur after a finite number of steps; that is, for some integer t :
qt1 = bqt + rt.
Then n = (rmrm1...r1r0)b.
NGUYEN CANH Nam Mathematics I - Chapter 5
Ri d fi l
RingsFields
Ri f i t
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Rings and fiels Ring of integers
Euclidean Algorithm
Presentation of integers
Algorithm for expansion of n by base b.
Example
Example
Represent 1397 by base 8.
1397 = 8.174 + 5
174 = 8.21 + 6
21 = 8.2 + 5
2 = 8.0 + 2
Hence, 1397 = (2565)8.
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Rings and fiels
Midterm examination
Time : Monday, 25th October 2010Duration : 1h30m
Knowledge : From chapter 1 to chapter 5
Only dictionary (hard copy) is allowed
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