I just started with Kirwan's "Complex Algebraic Curves", and on page $29$ the notion of a complex algebraic curve is defined:
Let $P(x,y)$ be a non-constant polynomial in two variables with complex coefficients and [no repeated factors]. Then the complex algebraic curve in $\Bbb C^2$ defined by $P(x,y)$ is
$$C = \{(x,y) \in \Bbb C^2: P(x,y) = 0\}$$
[The reason for the assumption in this definition that $P(x,y)$ has no repeated factors is the theorem called Hilbert's Nullstellensatz..]
The version stated in the book is that two polynomials have the same zero set if and only if they have the same irreducible factors.
I don't see why the Nullstellensatz is needed. I mean sure, we need one direction of it, but it is the trivial direction (if $P$ and $Q$ have the same irreducible factors, then they have the same zero set, and so define the same complex algebraic curve), but not the "significant" part of the theorem - the one where all the work is actually done.
Am I not seeing something?