1. ## Factoring a polynomial

Ive got a little problem with this. How do i factor a polynomial over a ternary field? I know how to do it in binary...

in binary: x^7 - 1 = x^7 - 1 = (x+1)(x^3 + x + 1)(x^3 + x^2 + 1)

i get that

and now the question is how do i do it for ternary field

x^5 - 1 = ???

thx a lot for help

2. First in any ring you can write
$x^5-1=(x-1)(x^4+x^3+x^2+1)$

Confince yourself that $x^4+x^3+x^2+1$ has no linear factor by checking that this polynom has no ternary root.

It mean either this polynomial can't be factor anymore or
$x^4+x^3+x^2+1=(x^2+Ax+B)(x^2+C+D)$
Since BD=1 then B=D=1 or B+D=-1.

Now you can playing around with A,C and see whether we can find a suitable A and C.

3. one more thing...do any special rules apply like in the binary case? For example in the binary field i could say that x + x = 0...

can i do the same for ternary field?