Friday, June 28, 2019

abstract algebra - What is $(x^7-x)mod(x^6-3)$ equal to?



I'm trying to use Rabins test for irreducibility over finite fields , but in part of the test you need to calculate $gcd(f,x^{p^{n_i}}-xmodf)$ where in my case p=7 and n=6,3,2 as I'm testing if $f(x)=x^6-3$ is irreducible over GF(7).



My trouble is I don't know how to calculate this modulo, I know how to do it for integers and I know that in my case it implies that $x^6=3$. But after this i'm stuck.



could anyone work me through how to find what $(x^7-x)mod(x^6-3)$ is equal to ?



Also is Rabins test a good go to for testing if a polynomial is irreducible over a finite field ? Or is there perhaps less cumbersome methods for higher degree's of f(x) where degree f(x)>3 and so doesn't strictly need to be factored into linear polynomials in order to be irreducible ? (just suggestions would suffice )



Answer



Division algorithm:



$$x^7 - x = (x^6 - 3) (x) + (2x)$$



and this is valid because $\deg (2x) < \deg (x^6 - 3)$



So the remainder is $2x$.


No comments:

Post a Comment

analysis - Injection, making bijection

I have injection $f \colon A \rightarrow B$ and I want to get bijection. Can I just resting codomain to $f(A)$? I know that every function i...