I'm thinking about a problem in Ravi Vakil's algebraic geometry notes http://math.stanford.edu/~vakil/216blog/ (Exercise 3.2B) and I'm having trouble with the second part of the exercise as my understanding of complex analysis is quite rudimentary.
If to every open set $U\subset \mathbb{C}$ we associate the ring of holomorphic functions which admit a holomorphic square root, this defines a presheaf on $\mathbb{C}$ (I can see this).
Apparently however this is not a sheaf -- supposedly it fails to satisfy the gluing property.
To prove this, we need to take an open cover $\{U_i\}$ of some open $U\subseteq \mathbb{C}$, a holomorphic function $f_i:U_i\to \mathbb{C}$ for each $i$ which has a holomorphic square root (i.e. there exists a holomorphic function $g_i$ for each $i$ such that $f_i(z)=g_i(z)^2$ for all $z\in U_i$), which all agree nicely on the intersections of the various open sets, but such that it is impossible to find a holomorphic function $f:U\to \mathbb{C}$ with a holomorphic function which restricts to $f_i$ on each $U_i$.
Any help would be great -- my problem is in showing that such a function cannot exist.