# Thread: Splitting field over Z_2

1. ## Splitting field over Z_2

Determine the degree [K:F] of the extension K over F, where K is a splitting field for the given polynomial:

i) $F = Z_2, x^7 -1$
ii) $F = Z_2, x^7 -1$

I can do these type of problems for the case F=Q, but how do you go about it with a finite field such as Z_2?

2. Originally Posted by Amanda1990
Determine the degree [K:F] of the extension K over F, where K is a splitting field for the given polynomial:

i) $F = Z_2, x^7 -1$
ii) $F = Z_2, x^7 -1$

I can do these type of problems for the case F=Q, but how do you go about it with a finite field such as Z_2?
The splitting field of $x^7 - 1$ over $F$ is $K = F(\zeta)$ where $\zeta$ is a primitive $7$-th root of unity (we know such a primitive root exists because $2\not | 7$ where $2=\text{char}(F)$). The polynomial $x^7 - 1$ can be factored into irreducibles as $(x-1)(x^3+x+1)(x^3 + x^2+1)$. Therefore, $\zeta$ has degree $3$ over $F$ which means $[K:F]=3$.

There happens to be a more general theorem about cyclotomic extensions of finite fields but I do not want to say it because I think it will just lead to your confusion, I think it would be easier for you to try to understand the solution in the above paragraph.