So, it really bothers me, that I have to resort to geometrical reasoning (not that it's not insightful, but it just doesn't feel nice) for defining the n'th root of complex numbers and thus also to proof that every complex number casually has a n'th root.
I guess, I could define the n'th root of z as "the complex number, that multiplied n times with itself equals z", but I wouldn't know how to proof the existence from that very definition and also it just doesn't feel very rigorous or pleasing.
I was thinking of something akin to defining the root of a real number r as sup{x in R : x^n < r}, from which one can (more or less) easily proof that the n'th power of this number actually equals r.
Is there some analogous way of doing it for the complex numbers?
If you accept the idea that multiplication by a complex is a similarity transformation, where the angles add up, then the root is obtained by angle subdivision (and ordinary root of the modulus).