- #1

PsychonautQQ

- 784

- 10

## Homework Statement

Find the splitting field of x^9-1 over F_13 (the field of 13 elements)

## Homework Equations

## The Attempt at a Solution

Every element in the cyclic group F_13* will have order 13 since 13 is prime, and thus 1 is the only root of x^9-1 in F_13. Thus I did the long vision of Dividing (x^9-1)/(x-1) = x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1 and thus in F_13 x^9 - 1 = (x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1)(x-1)

Thus I was thinking that adjoining a primitive 9th root of unity to F_13 would allow x^9-1 to split completely because if c is a primitive nth root of unity then c^n-1 + c^n-2 + ... + c + 1 = 0, but then I was thinking that this is only true for if n is a prime number, and in our case 9 would be 9 so it would not be true. Now that I am typing this I am quite sure that adjoining a primitive ninth root of unity would not do the job since it's minimal polynomial (over Q at least) would be the 9th cyclotomic polynomial which is not x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1. Any of you PF genius's have advice? :D