Why we can not divide $\Theta =70$ degrees into seven angles $\Theta /7=10$ degrees using euclids tools. How I apply chebyshev polynomial to solve it.
Does any one can help, I am not sure how to solve this.
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The chebyshev polynomial will give you the cosine of the 70-degree angle as a polynomial of degree 7 in the cosine of the 10-degree angle. To divide the angle is equivalent to constructing the cosine of 10 degrees, given the cosine of 70 degrees, so it's equivalent to constructing a solution of a polynomial equation of degree 7. The polynomial is irreducible over the rationals, so you'd have to construct a number of degree 7 over the rationals. The Euclidean tools only permit you to draw lines and circles, and these can only lead to numbers of degree 1, 2, 4, 8, and so on, but not 7.
That's a sketch of the answer to your question. The details involve learning a fair bit of the theory of fields, which I commend to your attention.