Let F be a field. (a) Prove that the polynomials a(x, y) = x^2 − y^2,...

Let F be a field.

(a) Prove that the polynomials a(x, y) = x^2 − y^2, b(x, y) = 2xy and c(x, y) = x^2 + y^2 in F[x, y] form a Pythagorean triple. That is, a^2 + b^2 = c^2. Use this fact to explain how to generate right triangles with integer side lengths.
(b) Prove that the polynomials a(x,y) = x^2 − y^2, b(x,y) = 2xy − y^2 and c(x,y) = x^2 − xy + y2 in F[x,y] satisfy the equation a^2 − ab + b^2 = c^2. Use this fact to explain how to generate triangles with integer side lengths containing

an angle of π /3
(c) Explain how to generate triangles with integer side lengths containing an angle of 2π / 3

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Expert Solution

Colby Messinger answered 1 year ago

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