I'm a little bit confused on how to factor an expression containing a complex number . To elucidate my concerns I shall give an example ;
Say we have $x^4-z$, where z is a complex number can we factor this as ,
$x^4-z=(x+i\sqrt[4]{z})(x-i\sqrt[4]{z})(x+\sqrt[4]{z})(x-\sqrt[4]{z})$ ?
Edit: I have edited this post to fix my mistake, originally i had put z where $\sqrt[4]{z}$ should have been
The expression $\sqrt[4]{z}$ is at best ambiguous, at worst meaningless.
To factor $$ x^4 - z $$ in the complex numbers you need to find the four fourth roots of $z$. The most straightforward way to do that is to find one of them by finding the fourth root of the modulus and one fourth of the argument. (This is what @SimplyBeautifulArt says in his comment.) For example, one fourth root of $i$ is $$ w = e^{\pi/8} = \cos(\pi/8) + i \sin(\pi/8). $$ Then you find all four fourth roots by multiplying by the fourth roots of $1$. Those are $\pm 1$ and $\pm i$.
So (as @deyore comments) $$ x^4 - i = (x-w)(x+w)(x-iw)(x+iw). $$
No one of those fourth roots is reliably singled out by $\sqrt[4]{i}$ so you should avoid that notation. (Sometimes by convention the fourth root in the first quadrant could be so labelled, but please don't.)