### All Abstract Algebra Resources

## Example Questions

### Example Question #1 : Abstract Algebra

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

**Possible Answers:**

**Correct answer:**

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:**

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

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

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

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

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

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

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

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

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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