week21 - More on Constructible Numbers and Angles Original...

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More on Constructible Numbers and Angles Original Notes adopted from March 5, 2002 (Week 21) © P. Rosenthal , MAT246Y1, University of Toronto, Department of Mathematics typed by A. Ku Ong Lemma : If x 0 is a root of a polynomial with coefficients in F(r), then x 0 is a root of a polynomial with coefficients in F. Proof: Given (a n + b n r) x 0 n + (a n-1 + b n-1 r) x 0 n-1 + . .. +(a 1 + b 1 r)x 0 + a 0 + b 0 r =0 with a i , b i F. By Dividing through by a n + b n r, we can assume it is poly monic. (ie. coefficients of x 0 n is 1). x 0 n (a n-1 + b n-1 r) x 0 n-1 + ... +(a 1 + b 1 r)x 0 + a 0 + b 0 r =0 Then, x 0 n + a n-1 x 0 n-1 +.... + a 1 x 0 + a 0 = r (b n-1 x 0 n-1 + ... + b 1 x 0 + b 0 ) Square both Sides (x 0 n + a n-1 x 0 n-1 +.... + a 1 x 0 + a 0 ) 2 - r (b n-1 x 0 n-1 + ... + b 1 x 0 + b 0 ) 2 =0 Recall: x 0 algebraic if p(x 0 )= 0 some polynomial p with rational coefficients. Theorem: Every constructible number is algebraic. Proof: Suppose x 0 is constructible. There exists Q = F 0 F 1 F 2 ...... F k with F i = F i-1 ( r i ) some r i F i-1 , r i = 0, r i F i-1 such that x 0 F
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week21 - More on Constructible Numbers and Angles Original...

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