Abstract Algebra : Abstract Algebra

Study concepts, example questions & explanations for Abstract Algebra

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Example Questions

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Example Question #1 : Abstract Algebra

Which of the following is an identity element of the binary operation ?

Possible Answers:

Correct answer:

Explanation:

Defining the binary operation  will help in understanding the identity element. Say  is a set and the binary operator is defined as  for all given pairs in .

Then there exists an identity element  in  such that given,

Therefore, looking at the possible answer selections the correct answer is,

Example Question #1 : Introduction

Which of the following illustrates the inverse element?

Possible Answers:

Correct answer:

Explanation:

For every element in a set, there exists another element that when they are multiplied together results in the identity element.

In mathematical terms this is stated as follows.

For every  such that  where  and  is an identity element.

Example Question #1 : Abstract Algebra

identify the following definition. 

Given  is a normal subgroup of , it is denoted that  when the group of left cosets of  in  is called __________.

Possible Answers:

Subgroup

Cosets

Simple Group

Normal Group

Factor Group

Correct answer:

Factor Group

Explanation:

By definition of a factor group it is stated,

Given  is a normal subgroup of , it is denoted that  when the group of left cosets of  in  is called the factor group of  which is determined by .

Example Question #4 : Abstract Algebra

Determine whether the statement is true of false:

Possible Answers:

True

False

Correct answer:

True

Explanation:

This statement is true based on the following theorem.

For all  in .

If  is a normal subgroup of  then the cosets of  forms a group under the multiplication given by,

Example Question #1 : Abstract Algebra

Which of the following is an ideal of a ring?

Possible Answers:

Minimum Ideal

All are ideals of rings.

Prime Ideal

Multiplicative Ideal

Associative Ideal

Correct answer:

Prime Ideal

Explanation:

When dealing with rings there are three main ideals 

Proper Ideal: When  is a commutative ring, and  is a non empty subset of  then,  is said to have a proper ideal if both the following are true.

 and 

Prime Ideal: When  is a commutative ring,  is a prime ideal if 

 is true and  

Maximal Ideal: When  is a commutative ring, and  is a non empty subset of  then,  has a maximal ideal if all ideal  are 

Looking at the possible answer selections, Prime Ideal is the correct answer choice.

Example Question #6 : Abstract Algebra

Which of the following is an ideal of a ring?

Possible Answers:

Associative Ideal

None are ideals

Maximal Ideal

Communicative Ideal

Minimal Ideal

Correct answer:

Maximal Ideal

Explanation:

When dealing with rings there are three main ideals 

Proper Ideal: When  is a commutative ring, and  is a non empty subset of  then,  is said to have a proper ideal if both the following are true.

 and 

Prime Ideal: When  is a commutative ring,  is a prime ideal if 

 is true and  

Maximal Ideal: When  is a commutative ring, and  is a non empty subset of  then,  has a maximal ideal if all ideal  are 

Looking at the possible answer selections, Maximal Ideal is the correct answer choice.

Example Question #7 : Abstract Algebra

Which of the following is an ideal of a ring?

Possible Answers:

Proper Ideal

Associative Ideal

Minimal Ideal

Communicative Ideal

All are ideals

Correct answer:

Proper Ideal

Explanation:

When dealing with rings there are three main ideals 

Proper Ideal: When  is a commutative ring, and  is a non empty subset of  then,  is said to have a proper ideal if both the following are true.

 and 

Prime Ideal: When  is a commutative ring,  is a prime ideal if 

 is true and  

Maximal Ideal: When  is a commutative ring, and  is a non empty subset of  then,  has a maximal ideal if all ideal  are 

Looking at the possible answer selections, Prime Ideal is the correct answer choice.

Example Question #2 : Abstract Algebra

What definition does the following correlate to?

If  is a prime, then the following polynomial is irreducible over the field of rational numbers.

Possible Answers:

Ideals Theorem

Primitive Field Theorem

Eisenstein's Irreducibility Criterion

Gauss's Lemma

Principal Ideal Domain

Correct answer:

Eisenstein's Irreducibility Criterion

Explanation:

The Eisenstein's Irreducibility Criterion is the theorem for which the given statement is a corollary to.

The Eisenstein's Irreducibility Criterion is as follows.

is a polynomial with coefficients that are integers. If there is a prime number  that satisfy the following,

Then over the field of rational numbers  is said to be irreducible. 

Example Question #1 : Abstract Algebra

Identify the following definition.

For some subfield of , in the Euclidean plane , the set of all points  that belong to that said subfield is called the __________.

Possible Answers:

Constructible Line

Angle

Plane

None of the answers.

Line

Correct answer:

Plane

Explanation:

By definition, when  is a subfield of , in the Euclidean plane , the set of all points  that belong to  is called the plane of .

Example Question #10 : Abstract Algebra

Identify the following definition.

Given that  lives in the Euclidean plane . Elements , and  in the subfield  that form a straight line who's equation form is , is known as a__________.

Possible Answers:

Line in 

Subfield

Plane

Angle

Circle in 

Correct answer:

Line in 

Explanation:

By definition, given that  lives in the Euclidean plane . When elements , and  in the subfield  , form a straight line who's equation form is , is known as a line in .

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