- #1

kmitza

- 17

- 4

- TL;DR Summary
- I have been revising some calculation skills in Galois theory and I ran into a problem. if we have an irreducible polynomial of degree 3 over GF(3) and we construct GF(27) as GF(3)/<f> and we have another such polynomial g(x) how do we find it's explicit roots in GF(27)

I am going to give up a bit more on the given problem. We start with polynomial ## x^27 -x ## over GF(3)[x] and we factorize it using a well known theorem it turns out it factorises into the product of monic polynomials of degree 1 and 3, 11 of them all together.

We then choose one of those polynomials and consider ## GF(3)/<x^3+x^2+2> ## and we're asked to find roots for the other 7 irreducible polynomials of degree 3 in GF(27) constructed this way and give isomorphisms between ## GF(3)/<f> ## and ## GF(3)/<g> ## for each of them.

It's not hard to see how to give the isomorphism but I am struggling with finding the explicit roots. I am wondering if anyone knows of a good method for doing this that isn't just plugging in all 27 elements the first time and then 24 and so on...

I know that there are probably computer methods for doing this but I am not looking for those.

We then choose one of those polynomials and consider ## GF(3)/<x^3+x^2+2> ## and we're asked to find roots for the other 7 irreducible polynomials of degree 3 in GF(27) constructed this way and give isomorphisms between ## GF(3)/<f> ## and ## GF(3)/<g> ## for each of them.

It's not hard to see how to give the isomorphism but I am struggling with finding the explicit roots. I am wondering if anyone knows of a good method for doing this that isn't just plugging in all 27 elements the first time and then 24 and so on...

I know that there are probably computer methods for doing this but I am not looking for those.