Solution Manual for First Course in Abstract Algebra A, 8th
Edition by John B. Fraleigh, Verified Chapters 1 - 56, Complete
Newest Version
The set {0, ±1, ±2,...} of ________ will be denoted by Z. - ANSWER:Integers
We will use N for the set {0,1,2,...} of _______ ________. - ANSWER:Natural numbers
An integer a is called a ________ of an integer b if a=bq for some integer q. In this
case we also say that b is a _______ of a, and we will use the notation b|a.
b is a ______ of a.
a is _________ of b. - ANSWER:Multiple
Divisor
Factor
Divisible
Every nonempty set of natural numbers contains a smallest element - ANSWER:Well-
Ordering Principle
For any integers a and b, with b>0, there exist unique integers q (quotient) and r
(remainder) such that a = bq+r, with 0≤r<b - ANSWER:Division algorithm
A set of multiples aZ has the property that the sum or difference of. two integers in
the set is again in the set, since aq₁±aq₂ = a(q₁±q₂). We say that the set aZ is ______
_____ ________ ___ ___________. - ANSWER:Closed under addition and subtraction
Let a and b be integers, not both zero. A positive integer d is called the ________
______ _______ of a and b if
(i) d is a divisor of a and b, and
(ii) any divisor of both a and b is also a divisor of d
denoted gcd(a,b) or (a,b). - ANSWER:Greatest common divisor
If a and b are integers, then we will refer to any integer of the form ma+nb, where
n∈Z, as a ______ ___________ of a and b. - ANSWER:Linear combination
Given integers a>b>0, the _________ _________ uses the division algorithm
repeatedly to obtain
a=bq₁+r₁ with 0≤r₁<b
b=r₁q₂+r₂ with 0≤r₂<r₁
r₁q₃+r₃ with 0≤r₃<r₂
etc. - ANSWER:Euclidean algorithm
, A number n is called _______ if it is equal to the sum of its proper positive divisor
(those divisors different from n). - ANSWER:Perfect
The nonzero integers a and b are said to be __________ _____ if (a,b)=1 -
ANSWER:Relatively prime
An integer p>1 is called a _____ ______ if its only divisors are ±1 and ±p.
An integer a>1 is called _________ if it is not prime. - ANSWER:Prime number
Composite
The _____ __ ____________ is an ancient algorithm for finding all prime numbers up
to any given limit. - ANSWER:Sieve of Eratosthenes
Any integer a>1 can be factored uniquely as a product of prime numbers in the form
a = P₁¹P₂²...Pₙⁿ where P₁<P₂...<Pₙ and the exponents α₁,α₂,...,αₙ are all positive. -
ANSWER:Fundamental Theorem of Arithmetic
There exist infinitely many prime numbers - ANSWER:Euclid Theorem
A positive integer m is called the _____ ______ ________ of the nonzero integers a
and b if
(i) m is a multiple of both a and b, and
(ii) any multiple of both a and b is also a multiple of m
We will use the notation lcm[a,b] or [a,b] for the _____ ______ _______ of a and b -
ANSWER:Least common multiple
A positive integer a is called a ______ if a=n² for some n∈Z. - ANSWER:Square
A positive integer is called ______-____ if it is a product of distinct primes. -
ANSWER:Square-free
If a,b,c are positive integers such that a²+b²=c², then (a,b,c) is called a ___________
_____. - ANSWER:Pythagorean triple
Let n be a positive integer. Integers a and b are said to be _________ ______ n if
they have the same remainder when divided by n.. This is denoted by writing
a≡b(mod n). - ANSWER:Congruent modulo
When working with congruence modulo n, the integer n is called _______. -
ANSWER:Modulus
Let n and m be positive integers, with (n,m)=1. Then the system of congruences
x≡a(mod n) x≡b(mod n)
has a solution. Moreover, any two solutions are congruent modulo mn. -
ANSWER:Chinese Remainder Theorem
Edition by John B. Fraleigh, Verified Chapters 1 - 56, Complete
Newest Version
The set {0, ±1, ±2,...} of ________ will be denoted by Z. - ANSWER:Integers
We will use N for the set {0,1,2,...} of _______ ________. - ANSWER:Natural numbers
An integer a is called a ________ of an integer b if a=bq for some integer q. In this
case we also say that b is a _______ of a, and we will use the notation b|a.
b is a ______ of a.
a is _________ of b. - ANSWER:Multiple
Divisor
Factor
Divisible
Every nonempty set of natural numbers contains a smallest element - ANSWER:Well-
Ordering Principle
For any integers a and b, with b>0, there exist unique integers q (quotient) and r
(remainder) such that a = bq+r, with 0≤r<b - ANSWER:Division algorithm
A set of multiples aZ has the property that the sum or difference of. two integers in
the set is again in the set, since aq₁±aq₂ = a(q₁±q₂). We say that the set aZ is ______
_____ ________ ___ ___________. - ANSWER:Closed under addition and subtraction
Let a and b be integers, not both zero. A positive integer d is called the ________
______ _______ of a and b if
(i) d is a divisor of a and b, and
(ii) any divisor of both a and b is also a divisor of d
denoted gcd(a,b) or (a,b). - ANSWER:Greatest common divisor
If a and b are integers, then we will refer to any integer of the form ma+nb, where
n∈Z, as a ______ ___________ of a and b. - ANSWER:Linear combination
Given integers a>b>0, the _________ _________ uses the division algorithm
repeatedly to obtain
a=bq₁+r₁ with 0≤r₁<b
b=r₁q₂+r₂ with 0≤r₂<r₁
r₁q₃+r₃ with 0≤r₃<r₂
etc. - ANSWER:Euclidean algorithm
, A number n is called _______ if it is equal to the sum of its proper positive divisor
(those divisors different from n). - ANSWER:Perfect
The nonzero integers a and b are said to be __________ _____ if (a,b)=1 -
ANSWER:Relatively prime
An integer p>1 is called a _____ ______ if its only divisors are ±1 and ±p.
An integer a>1 is called _________ if it is not prime. - ANSWER:Prime number
Composite
The _____ __ ____________ is an ancient algorithm for finding all prime numbers up
to any given limit. - ANSWER:Sieve of Eratosthenes
Any integer a>1 can be factored uniquely as a product of prime numbers in the form
a = P₁¹P₂²...Pₙⁿ where P₁<P₂...<Pₙ and the exponents α₁,α₂,...,αₙ are all positive. -
ANSWER:Fundamental Theorem of Arithmetic
There exist infinitely many prime numbers - ANSWER:Euclid Theorem
A positive integer m is called the _____ ______ ________ of the nonzero integers a
and b if
(i) m is a multiple of both a and b, and
(ii) any multiple of both a and b is also a multiple of m
We will use the notation lcm[a,b] or [a,b] for the _____ ______ _______ of a and b -
ANSWER:Least common multiple
A positive integer a is called a ______ if a=n² for some n∈Z. - ANSWER:Square
A positive integer is called ______-____ if it is a product of distinct primes. -
ANSWER:Square-free
If a,b,c are positive integers such that a²+b²=c², then (a,b,c) is called a ___________
_____. - ANSWER:Pythagorean triple
Let n be a positive integer. Integers a and b are said to be _________ ______ n if
they have the same remainder when divided by n.. This is denoted by writing
a≡b(mod n). - ANSWER:Congruent modulo
When working with congruence modulo n, the integer n is called _______. -
ANSWER:Modulus
Let n and m be positive integers, with (n,m)=1. Then the system of congruences
x≡a(mod n) x≡b(mod n)
has a solution. Moreover, any two solutions are congruent modulo mn. -
ANSWER:Chinese Remainder Theorem
