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Abstract Algebra : Fields

Study concepts, example questions & explanations for Abstract Algebra

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Example Questions

Example Question #1 : Splitting Fields

What definition does the following correlate to?

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

$\phi(x)=x^{p-1}+...+x+1$

Possible Answers:

Gauss's Lemma

Principal Ideal Domain

Ideals Theorem

Primitive Field Theorem

Eisenstein's Irreducibility Criterion

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.

$f(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_0$

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

$\\a_{n-1}\equiv a_{n-2}\equiv ...\equiv a+0\equiv 0 (\mod b)) \\a_n\not\equiv 0 (\mod b), a_0\not\equiv 0 (\mod b^2)$

Then over the field of rational numbers f(x) is said to be irreducible.

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Example Question #1 : Fields

Identify the following definition.

For some subfield of $\mathbb{R}$ , in the Euclidean plane $\mathbb{R}^2$ , the set of all points (x,y) that belong to that said subfield is called the __________.

Possible Answers:

Constructible Line

Plane

Line

None of the answers.

Angle

Correct answer:

Plane

Explanation:

By definition, when is a subfield of $\mathbb{R}$ , in the Euclidean plane $\mathbb{R}^2$ , the set of all points (x,y) that belong to is called the plane of .

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Example Question #2 : Fields

Identify the following definition.

Given that lives in the Euclidean plane $\mathbb{R}^2$ . Elements , , and in the subfield that form a straight line who's equation form is ax+by+c=0 , is known as a__________.

Possible Answers:

Plane

Subfield

Line in

Angle

Circle in

Correct answer:

Line in

Explanation:

By definition, given that lives in the Euclidean plane $\mathbb{R}^2$ . When elements , , and in the subfield , form a straight line who's equation form is ax+by+c=0 , is known as a line in .

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Example Question #11 : Abstract Algebra

Identify the following definition.

Given that lives in the Euclidean plane $\mathbb{R}^2$ . Elements , , and in the subfield that form a straight line who's equation form is ax+by+c=0 , is known as a__________.

Possible Answers:

Plane

Subfield

Line in

Angle

Circle in

Correct answer:

Line in

Explanation:

By definition, given that lives in the Euclidean plane $\mathbb{R}^2$ . When elements , , and in the subfield , form a straight line who's equation form is ax+by+c=0 , is known as a line in .

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Example Question #1 : Geometric Fields

Identify the following definition.

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

Possible Answers:

Constructible Number

Magnitude

Angle

Plane

Straight Line

Correct answer:

Constructible Number

Explanation:

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

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Abstract Algebra : Fields

Study concepts, example questions & explanations for Abstract Algebra

All Abstract Algebra Resources

Example Questions

Example Question #1 : Splitting Fields

Example Question #1 : Fields

Example Question #2 : Fields

Example Question #11 : Abstract Algebra

Example Question #1 : Geometric Fields

All Abstract Algebra Resources

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