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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 04/26/2011 for the course MATH 246 taught by Professor Applebaugh during the Spring '10 term at University of Toronto.

### Page1 / 4

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online