Fix $N>0$. Let $b\_i=(b\_{i,1}, b\_{i,2}, b\_{i,3}, b\_{i,4})$, $i=1,\ldots, m$, be distinct 4-tuples of integers with with all $0\leq b\_{i,j}< N$. (The zero tuple is disallowed.) 


Define $w\_i=(\prod\_{j=1}^4 (z-z\_j)^{b\_{i,j}})^{\frac1{N}}$. 


Consider $w\_i$ as an element of the following vector space: the algebraic closure of $\mathbb{C}(z)$ over the field $\mathbb{C}(z)$. 


I believe that Puiseux series can be used to show the linear independence of the $w\_i$. Are there any other approaches that might show the linear independence of the $w\_i$? 


More generally, are there any other general techniques for proving linear independence of functions in the given vector space? 



Linear independence in the algebraic closure of $\mathbb{C}(z)$
You can also use Galois theory or monodromy. Take a minimal linear dependence relation and apply the automorphimsm of the algebraic closure that fixes $(z-z\_j)^{1/N},j>1$ and fot $j=1$ multiplies the function by an N-th root of unity, thus getting a new relation and you can produce a shorter relation from those two. 


