
roots of i, polynomial
Let alpha and alpha be the square roots of i.
Write down a degree two polynomial having alpha and alpha as its roots. Then write down a degree four polynomial with integer coefficients having alpha and negative alpha among its roots. What are the other two roots of this polynomial.
I don't know if I can't do this because I really don't understand what the question is asking me or if the question is just going straight over my head :(

What about $\displaystyle \displaystyle f(z) = (z  \alpha)(z + \alpha)$?

I'm really sorry. I still don't get where to go.

The value $\displaystyle \alpha=\cos\left(\frac{\pi}{4}\right)+i~\sin\left( \frac{\pi}{4}\right) $, that is one square root of $\displaystyle \mathif{i}$.
Is $\displaystyle \alpha$ a root of $\displaystyle z^4+1=0~?$
What are the other roots?