I need to find cyclotomic cosets depending on $n=7$ and $q=4$ and find the factorization of $x^7-1$ into irreducible factors over $GF(4)$.
Thanks for any advice.
I need to find cyclotomic cosets depending on $n=7$ and $q=4$ and find the factorization of $x^7-1$ into irreducible factors over $GF(4)$.
Thanks for any advice.
As has been noted by Jack D'Aurizio in his comment, the polynomial $x^{7}-1$ splits into a product of $x-1$ and two different irreducible factors of degree $3$ over $F_{2}.$ This certainly gives the same factorization (but not a priori into irreducible factors) over $F_{4}.$ However $F_{4}$ and $F_{16}$ contain no element of multiplicative order $7,$ so contain no root of $x^{7}-1$ other than $1,$ so the two factors of degree $3$ remain irreducible in $F_{4}[x].$