# Classical Impossible Constructions of Geometry proofs with Abstract Algebra.

• Trisecting an angle (dividing a given angle into three equal angles),
• Squaring a circle (constructing a square with the same area as a given circle), and
• Doubling a cube (constructing a cube with twice the volume of a given cube).

Told that these problems could only be proved with abstract algebra. I have no idea how to start. I have found this page.

I have an idea of what is being said, but no idea about how to exactly prove this. Any pointers would be helpful.

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Did you read Wikipedia? – lhf May 2 '12 at 11:54
@lhf yes, but I have, but I have having trouble how to "say it" with abstract algebra. – yiyi May 2 '12 at 11:58
@MaoYiyi: Do you know any Galois theory? If not, that's where to start. – Zhen Lin May 2 '12 at 12:03
@ZhenLin, no need for Galois theory, just field theory. Except perhaps for proving that you cannot square the circle, i.e., that $\pi$ is transcendental. Hadlock solves the other two problems right at the start of the book. – lhf May 2 '12 at 12:06
@ZhenLin no idea about that, just started learning abstract algebra. – yiyi May 2 '12 at 12:08

## 1 Answer

Field theory and its classical problems by Hadlock is a wonderful book motivated by these problems.

See also Geometry: Euclid and Beyond by Hartshorne.

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Thanks, this is what I was looking for, something to help me learn how to "say it" with abstract algebra. – yiyi May 2 '12 at 12:09
Possibly the most elementary book I know of for this (much more elementary than Hadlock's and Hartshorne's books) is Benjamin Bold, Famous Problems of Geometry and How to Solve Them, Dover Publications, 1982. See also the references at the end of pballew.net/Constructable_17gon.pdf Quite a few of these references are freely available on-line at the URL's provided. – Dave L. Renfro May 2 '12 at 15:53
@DaveL.Renfro you rock! thanks for the link. – yiyi May 3 '12 at 1:51
@DaveL.Renfro, thanks for the reference to Bold's book. I didn't know that book. – lhf May 3 '12 at 1:55