Splitting Field of $t^{50} + 1$ over $Z_{101}$

Question

I don't know how to deal with this exercise:

• Find the splitting field of $t^{50} + 1$ over $Z_{101}$.

Firstly, I don't know how to check the irreducibility in this case, but neither how to go on in order to find such field.

Thank you for your help.

Answer 1

By Fermat's little theorem every non-zero $x\in\Bbb Z_{101}$ is a zero of $g=t^{100}-1$. As there are $100$ different such elements and $g$ has degree $100$ it follows that $$g=\prod_{0\ne x\in\Bbb Z_{101}}(t-x)$$ As $(t^{50}-1)(t^{50}+1)=g$ it follows that also $t^{50}+1$ splits into linear factors over $\Bbb Z_{101}$. Hence the splitting field of $t^{50}+1$ is already $\Bbb Z_{101}$.