GCD of Two Polynomials over a Finite Field

Definition of the greatest common divisor of two polynomials over a field F as the unique monic polynomial of greatest degree that divides both polynomials. Demonstration of how to use the Euclidean Algorithm to compute the gcd of two polynomials over a finite field, and to find a linear combination of the two polynomials that equals the gcd. Includes examples and concept checks.

[@pcstrom]

Thank you! Most understandable worked examples I've seen.

[@roydmalenga9868]

Ring Theory exam in 4 hours and polynomials is yhe last thing left to understand. I've been failing to get this. Finally understand. Thank you so much for this video 🙏

[@samuelbassey6806]

Thanks for sharing ma❤❤❤, it really help me in school. Thanks once again

[@helinafedorchuk2286]

Thank you so much, Katherine! Your video is a piece of gold, so helpful!

[@keshavm-hy9cn]

Thank you so much mam.🙏🙏

[@katherinekelm7439]

20:39 Small typo in last line; right hand side of the equation should say 4x(3x+1) + 0.

[@chandramathematicspoint8491]

What will be gcd of X²+3 and X³+6X+1 in Z_7? Please answer with explanation.

[@katherinekelm7439]

If you use the Euclidean Algorithm to divide x³+6x+1 by x²+3, the last non-zero remainder is 3x+1. Factoring out 3, we get 3(x+5), so the gcd is x+5. You can check that x³+6x+1 = (x+5)(x² + 2x + 3) and x²+3 = (x+5)(x+2).

[@chandramathematicspoint8491]

Thanks

[@irisflystam9134]

So, two questions about the slide at 3:00 .
1. Possible typo: Isn't (x^3+2x^2+1)/(x+1) = x^2+x-1 ?
2. Why was x+1 multiplied by 2 in x+1 = (2x+2)*2 ? Was it to somehow get no remainder after the last calculation? Wouldn't 2(2x+2) = 4x+4 ? I feel like I'm missing something...

[@katherinekelm7439]

1. All coefficients are taken modulo 3, so x^2 + x - 1 is equal to x^2 + x + 2.
2. Yes. Again, 4x+4 = x+1 over Z_3.

[@irisflystam9134]

@@katherinekelm7439 Right! Thank you! I missed the Z_3, I'm unused to finishing with a modulo...

[@pedromendes6846]

@@katherinekelm7439 So what would happen if we had a polynomial like 3x+3 over Z_3, would it all turn into 0 since 3 mod 3 = 0 or x+1 ?

[@katherinekelm7439]

@@pedromendes6846 Yes, the coefficients are actually congruence classes in Z_3, so 3x+3 is actually [3]x+[3] = [0]x+[0].