461hw3sol - Math 461 F Spring 2011 Homework 3 Solutions...

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Unformatted text preview: Math 461 F Spring 2011 Homework 3 Solutions Drew Armstrong 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. The symbol √ 3 represents two numbers and thus 1 + √ 3 represents two numbers (both nonzero). Each of these numbers in turn has two square roots, so the symbol p 1 + √ 3 represents four distinct numbers. Let’s look for an equation that these numbers must satisfy. Let x represent any value of the expression p 1 + √ 3. Then x = q 1 + √ 3 x 2 = 1 + √ 3 x 2- 1 = √ 3 ( x 2- 1) 2 = 3 x 4- 2 x 2- 2 = 0 . Note that this last equation has at most four solutions. Hence it has exactly four solutions: the numbers p 1 + √ 3. A.2. Use the trigonometric identity cos(3 θ ) = 4cos 3 θ- 3cos 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.) Put θ = π/ 9 into the equation cos(3 θ ) = 4cos 3 θ- 3cos θ to get the equation 1 / 2 = cos( π/ 3) = 4cos 3 ( π/ 9)- 3cos( π/ 9). Letting x = cos( π/ 9) we get 4 x 3- 3 x- 1 2 = 0 , or x 3- 3 4 x- 1 8 = 0 . Now we apply Cardano’s formula (page 5 in the text) to get cos( π/ 9) = x = 3 s 1 16 + r 1 16 2- 4 16 2- 3 s- 1 16 + r 1 16 2- 4 16 2 = 3 r 1 16 + 1 16 √- 3- 3 r- 1 16 + 1 16 √- 3 = 3 s 1 + i √ 3 16- 3 s- 1 + i √ 3 16 = 1 3 √ 16 3 q 1 + i √ 3 + 3 q 1- i √ 3 . Actually, this formula represents three different numbers, one of which is cos( π/ 9). A.3. Suppose that p = 2 a + 1 is a prime number. Show that a must be a power of 2....
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This note was uploaded on 01/08/2012 for the course MATH 561 taught by Professor Armstrong during the Spring '11 term at University of Miami.

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461hw3sol - Math 461 F Spring 2011 Homework 3 Solutions...

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