Answer to Question #301814 in Algebra for kavya

Question

1. Find the greatest common divisor of the following polynomials over F, the field of rational numbers: (a) x 3 - 6x 2 + x + 4 and x 5 - 6x + 1. (b) x 2 + 1 and x6 + x 3 + x + 1. 

Answer 1

(a)

$f(x)=x^3-6x^2+x+4\\ g(x)=x^5-6x+1\\ g:f\\ \begin{matrix} x^5-6x+1 &|x^3-6x^2+x+4 \\ x^5-6x^4+x^3+4x^2 & x^2+6x+35\\ ---------\\ 6x^4-x^3-4x^2-6x+1\\ 6x^4-36x^3+6x^2+24x\\ -----------\\ 35x^3-10x^2-30x+1\\ 35x^3-210x^2+35x+140\\ ----------\\ 200x^2-65x-139 \end{matrix}$

$r=200x^2-65x-139\\ f:r$

$\begin{matrix} x^3-6x^2+x+4 & |200x^2-65x-139 \\ ------- & x-1135\\ 200x^3-1200x^2+200x+800\\ 200x^3-65x^2-139x\\ --------\\ -1135x^2+339x+800\\ ---------\\ -22700x^2+67800x+160000\\ -22700x^2+73775x+157765\\ --------\\ -5975x+2235\\ ------\\ -1195x+447 \end{matrix}$

$r_1=-1195x+447\\ r:r_1\\ \begin{matrix} 200x^2-65x-139 &|-1195x+447 \\ -------& 200x+2345\\ -23900x^2+77675x+166105\\ -23900x^2+89400x\\ -------\\ -11725x+166105\\ --------\\ 2345x+33221\\ ------\\ -2802275x-39699095\\ -2802275x-1048215\\ -----\\ -38650880\neq0 \end{matrix}\\ gcd(f,g)=1$

(b)

$f(x)=x^6+x^3+x+1\\ g(x)=x^2+1\\ f:g\\ \begin{matrix} x^6+x^3+x+1& |x^2+1 \\ x^6+x^4 & x^4-x^2+x+1\\ -----\\ -x^4+x^3+x+1\\ -x^4-x^2\\ -----\\ x^3+x^2+x+1\\ x^3+x\\ -----\\ x^2+1\\ x^2+1\\ -----\\0 \end{matrix}\\ gcd(f,g)=x^2+1$