Ring Theory as Applied to the Three Classical Geometric Construction Problems

Ring Theory as Applied to the Three Classical Geometric Construction Problems

The Negative Proofs

Now to show that it is impossible to Circle the Sqaure, Double the Cube, and to Trisect an Arbitrary Angle. We just found that for any intermediate field between the rationals and C

[F(a₁,...,a_n):Q] = 2^m

must be true, but we know that

is transcendental, so [Q(

):Q] =

2^mtherefore it is impossible to square the circle!

Also, [Q(³):Q] = 3 2^mso it is impossible to double the cube!

Last we can use the trig identity:

cos(A) = 4 cos³(A/3) - 3 cos (A/3).

Setting A=60⁰ and x = cos(A/3) we get

1/2 = 4x³ - 3x
or
0 = 8 x³ - 6x -1

which tells us that [Q(x):Q] = 3

2^m. In other words it is impossible to construct the cosine of 20⁰. Therefore it is impossible to trisect the original angle, 60⁰. meaning, obviously, that it is impossible to trisect an arbitrary angle!

This concludes our little story on the three classical geometric construction problems. It is amazing how using the power of modern mathematics we can solve a puzzle which perplexed so many, for so long so ... simply!

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