 # 1. Use DeMoivre's Theorem to help you find a nonzero polynomial tha...

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1. Use DeMoivre's Theorem to help you find a nonzero polynomial that cos (60) = f(cos(()). 3. (a) Compute IQ(i, 3) : Q]. (b) Show that i E Q(i + 2V3). This will require some computations. (c) Show that V3 E Q(i + 2v3. (d) Show that Q(i, (3) = Q(i + 2v3) (e) Compute [Q(i+2v3::Q]. (f) Without finding the polynomial, what is the degree of the minimum polynomial for i + 2V3 over Q. Briefly explain. (g) Using your answer from part (f), find the minimum polynomial for i + 2V3 over Q. 4. Let f (x) = anx7 + - + + apx + ao € Q[x] and g(x) = xnf(t). . (a) Explain why g(x) is a polynomial and what is its degree? (b) What are the coefficients of each power of x in g(x) compared to the coefficients of f (x) ? (c) If a1, 02, , an E C are the roots of g(x), what are the roots of f(x)? (d) Is x7 - 15x5 + 21x + 3 solvable by radicals? You may apply any theorems from the textbook. (e) Apply your results from parts (c) and (d) to determine if 3.77 + 21.x - 15.x² + 1 is solvable by radicals. ?

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