How to find the gcd of f(x)=x^{2}- x-2 and g(x)= x^{3}-7x+6 in F_{3}[x]. Expressed as a linear combination of f,g.

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- May 4th 2013, 07:34 PMkayabi5Find gcd
How to find the gcd of f(x)=x

^{2}- x-2 and g(x)= x^{3}-7x+6 in F_{3}[x]. Expressed as a linear combination of f,g. - May 4th 2013, 08:22 PMibduttRe: Find gcd
factorize f(x) and g(x) .

the product of common factors will be the gcd. - May 5th 2013, 02:13 AMabualabedRe: Find gcdhello every one
- May 5th 2013, 05:18 PMkayabi5Re: Find gcd
Ok, but what throws me off is the F3[x]. I know its (0,1,2), but how to you apply it?

- May 6th 2013, 05:20 AMHallsofIvyRe: Find gcd
So we are dealing in modulo 3. The first thing I would do is simplify $\displaystyle x^3- 7x+ 6$: 7= 1 (mod 3) and 6= 0 (mod 3). So $\displaystyle x^3- 7x+ 6= x^2- x= x(x^2- 1)= x(x- 1)(x+ 1)$ which we could also write as x(x+2)(x+1). To factor $\displaystyle x^2- x- 2= x^2+ 2x+ 1 (mod 3)$, look at the general $\displaystyle (x+ a)(x+ b)= x^2+ (a+b)x+ ab$. We need to find a and b such that a+ b= 2 (mod 3) and ab= 1 (mod 3). a= b= 1 should be obvious.