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Q20E

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Abstract Algebra: An Introduction

Found in: Page 470

Book edition 3rd

Author(s) Thomas W Hungerford, David Leep

Pages 608 pages

ISBN 9781111569624

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Short Answer

Prove that the midpoint of the line segment between two constructible points is a constructible point. [Hint: Adapt the hint to Exercise $17$ .]

The midpoint of the segment between two constructible points is a constructible point.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Conceptual Introduction

Drawing precise lines, line segments, shapes, circles, and other figures with a ruler, compass, or protractor is known as geometric construction.

Step 2: Properties of constructible.

A point $(r, s)$ is constructible iff $r$ and $s$ are constructible.
If $a, b, c$ with $c \neq 0$ are constructible then $a + b, a - b$ and $a / c$ are constructible.

Step 3: Proving that the midpoint of the line segment between two constructible point is also constructible.

The proving part:

Let $ℚ [x]$ , $(a_{1}, a_{2})$ and $(b_{1}, b_{2})$ are constructible points.

Then from $1$ , each $a_{1}, a_{2}, b_{1}, b_{2}$ are constructible.

Form point $2$ in step 1^st, we infer that $\frac{a_{1} + b_{1}}{2}$ and $\frac{a_{2} + b_{2}}{2}$ are constructible numbers.

Then by reverse of point $1$ in step 1^st, the points $(\frac{a_{1} + b_{1}}{2}, \frac{a_{2} + b_{2}}{2})$ is constructible.

Hence, the midpoint of the line segment between two constructible points is a constructible point.

Most popular questions for Math Textbooks

Prove the converse of Theorem $15.6$ : If r is in some quadratic extension chain , then r is a constructible number.[Hint: Theorem $15.1$ and Corollary $15.2$ .)

Consider a rectangular box with a square bottom of edge and height . Assume the volume of the box is 3 cubic units and its surface area is 7 square units. Can the edges of this box be constructed with straightedge and compass?

Let $a, c$ be constructible numbers with $c \neq 0$ . Prove that $a / c$ is constructible. [Hint: The case when $a > 0, c > 0$ was done in the proof of Theorem $15.1$ .]

Prove that $c o s t = 4 c o s^{3} t - 3 c o s t$ .

Let F be a subfield of R and $k \in F$ . Show that $F (\sqrt{k}) = {a + b \sqrt{k} a, b \in F}$ is a subfield of C that contains F. If K>0 , show that F is a subfield of R.

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