Fields - Abstract Algebra

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Question

Identify the following definition.

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

Answer

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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Question

Identify the following definition.

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

Answer

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

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Question

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 , is known as a__________.

Answer

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 , is known as a line in .

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Question

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 , is known as a__________.

Answer

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 , is known as a line in .

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Question

What definition does the following correlate to?

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

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

Answer

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,

$\a_{n-1}\equiv a_{n-2}\equiv ...\equiv a+0\equiv 0 (\mod b)) \a_n ot\equiv 0 (\mod b), a_0 ot\equiv 0 (\mod b^2)$

Then over the field of rational numbers is said to be irreducible.

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