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Math 461 F Spring 2011 Homework 3 Drew Armstrong Reading. Section 2.4 Problems. A.1. How many diﬀerent (complex) numbers does the expression p 1 + √ 3 represent? Find a polynomial over Z which has these numbers as its roots. A.2. Use the trigonometric identity cos(3 θ ) = 4 cos 3 θ - 3 cos θ together with Cardano’s formula to ﬁnd an expression for cos( π/ 9). (Note: This expression must involve complex numbers because cos( π/ 9) is not constructible.) A.3. Suppose that p = 2 a + 1 is a prime number. Show that a must be a power of 2. (Hint: If a has an odd factor b , show that the polynomial x b +1 factors nicely.) A.4. Prove that √ 2 = 1 + 1 2 + 1 2 + 1 2 + . . . . (You can assume that the expression on the right converges.) We can de- scribe this process recursively by setting s 0 = 1 and s n = 1 + 1 / (1 + s n - 1 ) for n ≥ 1. What is s 4 ? How close is this to √ 2? Let D be the set of numbers that can be formed from
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