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)?
- 15x5 + 21x + 3 solvable by radicals? You may apply any theorems from the
(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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