May 2012
Beginner
160 pages
4h 24m
English
Content preview from Introduction to Abstract Algebra, Solutions Manual, 4th Edition
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6.5 Geometric Constructions
1.
a. Given the line through A and B and the point C, locate D on AB such that |AD| = |AC| . Then locate E such that |CE| = |DE|. The line through C and E is parallel to AB (ACED is a parallelogram).
2.
a. The line through (a, b) and (c, d) has equation 
c. Given ax + by = c and dx + ey = f in F[x], the solution (if any) involves only field operations in F, and so both x and y are in F (if they exist).
3. The question asks whether 
can be constructed. This is the result of bisecting 
. But the angle π/6 is constructible because 
is constructible (see diagram) so angle π/12 is constructible.
can be constructed. This is the result of bisecting . But the angle π/6 is constructible because is constructible (see diagram) so angle π/12 is constructible.
5. No. A sphere of radius 1 has volume 
and, if this is the volume of a cube with side a, then . But a is not constructible since it is not even algebraic over . For if ...
and, if this is the volume of a cube with side a, then . But a is not constructible since it is not even algebraic over . For if ...Become an O’Reilly member and get unlimited access to this title plus top books and audiobooks from O’Reilly and nearly 200 top publishers, thousands of courses curated by job role, 150+ live events each month,
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