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Abstract Algebra : Fields

Study concepts, example questions & explanations for Abstract Algebra

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Example Questions

Example Question #5 : Abstract Algebra

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.

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

Possible Answers:

Primitive Field Theorem

Principal Ideal Domain

Eisenstein's Irreducibility Criterion

Ideals Theorem

Gauss's Lemma

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.

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

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

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

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

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Example Question #6 : Abstract Algebra

Identify the following definition.

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

Possible Answers:

None of the answers.

Plane

Line

Constructible Line

Angle

Correct answer:

Plane

Explanation:

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

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Example Question #7 : Abstract Algebra

Identify the following definition.

Given that $\displaystyle A$ lives in the Euclidean plane $\mathbb{R}^2$ $\displaystyle \mathbb{R}^2$. 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 $\displaystyle A$

Line in $\displaystyle A$

Subfield

Angle

Plane

Correct answer:

Line in $\displaystyle A$

Explanation:

By definition, given that $\displaystyle A$ lives in the Euclidean plane $\mathbb{R}^2$ $\displaystyle \mathbb{R}^2$. 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 $\displaystyle A$.

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Example Question #11 : Abstract Algebra

Identify the following definition.

Possible Answers:

Line in $\displaystyle A$

Plane

Circle in $\displaystyle A$

Angle

Subfield

Correct answer:

Line in $\displaystyle A$

Explanation:

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Example Question #12 : Abstract Algebra

Identify the following definition.

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

Possible Answers:

Plane

Magnitude

Straight Line

Constructible Number

Angle

Correct answer:

Constructible Number

Explanation:

By definition if a line segment has length |a| $\displaystyle |a|$ 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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Abstract Algebra : Fields

Study concepts, example questions & explanations for Abstract Algebra

All Abstract Algebra Resources

Example Questions

Example Question #5 : Abstract Algebra

Example Question #6 : Abstract Algebra

Example Question #7 : Abstract Algebra

Example Question #11 : Abstract Algebra

Example Question #12 : Abstract Algebra

All Abstract Algebra Resources

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