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 \(\displaystyle b\) is a prime, then the following polynomial is irreducible over the field of rational numbers.

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

Possible Answers:

Gauss's Lemma

Ideals Theorem

Principal Ideal Domain

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.

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

is a polynomial with coefficients that are integers. If there is a prime number \(\displaystyle b\) that satisfy the following,

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

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

Example Question #1 : Geometric Fields

Identify the following definition.

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

Possible Answers:

Angle

Line

Constructible Line

None of the answers.

Plane

Correct answer:

Plane

Explanation:

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

Example Question #2 : Geometric Fields

Identify the following definition.

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

Possible Answers:

Circle in \(\displaystyle A\)

Subfield

Angle

Plane

Line in \(\displaystyle A\)

Correct answer:

Line in \(\displaystyle A\)

Explanation:

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

Example Question #3 : Geometric Fields

Identify the following definition.

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

Possible Answers:

Line in \(\displaystyle A\)

Subfield

Angle

Circle in \(\displaystyle A\)

Plane

Correct answer:

Line in \(\displaystyle A\)

Explanation:

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

Example Question #1 : Fields

Identify the following definition.

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

Possible Answers:

Constructible Number 

Angle

Plane

Magnitude

Straight Line

Correct answer:

Constructible Number 

Explanation:

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

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