Abstract Algebra : Fields

Study concepts, example questions & explanations for Abstract Algebra

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Example Questions

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 .

Example Question #1 : Fields

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:

Angle

Plane

Circle in 

Subfield

Line 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 .

Example Question #2 : Geometric Fields

Identify the following definition.

If a line segment has length  and is constructed using a straightedge and compass, then the real number  is a __________.

Possible Answers:

Angle

Constructible Number 

Plane

Magnitude

Straight Line

Correct answer:

Constructible Number 

Explanation:

By definition if a line segment has length  and it is constructed using a straightedge and compass then the real number  is a known as a constructible number.

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