I have the following exercise:
Find splitting field for the polynomial $x^2 + 1$ over $\mathbb{Z_3}$.
My solution:
At first, we should try to solve the equation $x^2 + 1 = 0$, thus $x^2 = 2$ and we need $\sqrt2$. Add this root to our new field and we have $\{0, 1, 2, \sqrt2, 2\sqrt2, 1+\sqrt2, 2 + \sqrt2, 1+2\sqrt2, 2 + 2\sqrt2 \}$ and that's our splitting field where roots of $x^2 + 1 = 0$ are $\sqrt2$ and $2\sqrt2$.
Is it correct or not? And I think there is no exact algorithm how to build a splitting field. How to do it properly?