8. [8 Marks] Let a, b, c, d be non-zero rational numbers, and let a = 1+√5 +√7-√√35. a) Can a¹ be written in the form a¹

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8 8 Marks Let A B C D Be Non Zero Rational Numbers And Let A 1 5 7 35 A Can A Be Written In The Form A 1 (37.69 KiB) Viewed 30 times

Abstract Algebra Ring theory
Please solve a, b, c and explain steps
8. [8 Marks] Let a, b, c, d be non-zero rational numbers, and let a = 1+√5 +√7-√√35. a) Can a¹ be written in the form a¹=e+f√5 + g√7+h√35 for rational numbers e, f, g, h? Fully justify your answer. b) You are told that a is a root of the polynomial p(x) = x³ +p₁2¹+P3x³ + p2r² + p1x + po with rational coefficients. Find three further roots. c) Find values of Po, P1, P2, P3, P4 which make this true.