461hw4 - 2 x-1 = 0 A.5 Prove that x 3 x 2-2 x-1 = 0 has no...

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Math 461 F Spring 2011 Homework 4 Drew Armstrong Reading. Section 2.4 Problems. A.1. Euclid’s Lemma. Suppose that a divides bc for a,b,c Z with a and b coprime (i.e. they have no common factor except ± 1). Prove that a must divide c . (Hint: Since a and b are coprime, you may assume — without proof — that there exist x,y Z such that ax + by = 1.) A.2. Prove that 3 2 is not rational. A.3. Consider a quadratic ﬁeld extension F F [ c ] = { a + b c : a,b F } and deﬁne the conjugation map a + b c 7→ a - b c . Prove that for all u,v F [ c ] we have u + v = u + v , uv = u v . A.4. Consider again the same ﬁeld extension F F [ c ] and let p ( x ) F [ x ] be a polynomial with coeﬃcients in F . Prove that for any α F [ c ] we have p ( α ) = 0 ⇐⇒ p ( α ) = 0 . For the next two problems you may assume — without proof — that 2 cos(2 π/ 7) is a root of x 3 + x 2 -
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Unformatted text preview: 2 x-1 = 0 . A.5. Prove that x 3 + x 2-2 x-1 = 0 has no rational root, and hence that cos(2 π/ 7) is not rational. A.6. Prove that cos(2 π/ 7) is not constructible, and hence that the regular heptagon is not constructible with straightedge and compass. Note: We have now proved that the following classical problems are im-possible: “doubling the cube”, “trisecting an angle”, “constructing the regular heptagon”. The only problem left is “squaring the circle”, which is equivalent to constructing π . Lindemann (1882) proved that π is not constructible, but I’m not clever enough to present the proof to you. (Wikipedia has it.)...
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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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