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  • 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](http://en.wikipedia.org/wiki/Compass_and_straightedge#Impossible_constructions)?2012-05-02
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    @lhf yes, but I have, but I have having trouble how to "say it" with abstract algebra.2012-05-02
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    @MaoYiyi: Do you know any Galois theory? If not, that's where to start.2012-05-02
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    @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.2012-05-02
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    @ZhenLin no idea about that, just started learning abstract algebra.2012-05-02
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    @lhf just started learning abstract algebra. Which book are you talking about?2012-05-02
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    @MaoYiyi, see my answer.2012-05-02

1 Answers 1

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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.2012-05-02
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    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 http://pballew.net/Constructable_17gon.pdf Quite a few of these references are freely available on-line at the URL's provided.2012-05-02
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    @DaveL.Renfro you rock! thanks for the link.2012-05-03
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    @DaveL.Renfro, thanks for the reference to Bold's book. I didn't know that book.2012-05-03