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  3. let f be a field. (a) prove that the polynomials

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

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Advance Math

Let F be a field.

  1. (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 generateright triangles with integer side lengths.

  2. (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 generatetriangles with integer side lengths containing

    an angle of ? /3

  3. (c) Explain how to generate triangles with integer side lengthscontaining an angle of 2? / 3

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