# Showing Elliptic Curves are irreducible

Suppose an elliptic curve defined over a field $K$ has form $y^2 + h(x)y = f(x)$ where $h(x), f(x) \in K[x]$. $h(x)$ has degree 0 or 1 and $f(x)$ has degree 3 or 4. Moreover, this equation is non-singular.

If characterstic of $K \neq 2$, show that the curve is absolutely irreducible.

-
non-singularity means there is no point on the curve that satisfies both partial derivatives. i.e. $2y + h(x) = 0$ and $h'(x)y = f'(x)$. – bonyankan Feb 13 '12 at 15:51
Welcome to MathSE. I see that you are relatively new here. So I wanted to let you know a few things about MathSE. We like to know where the problem is from what you've tried on a problem; this prevents people from wasting their time telling you thinks you already know, and helps make sure the answers are at an appropriate level. If this is homework, please consider adding the [homework] tag; people will still help, so don't worry. Also, posting questions in the imperative ("Compute", "Prove", "Show") is considered rude by some of the members, so please consider editing the post. Thank you. – Arturo Magidin Feb 13 '12 at 17:11
That's a very gentlemanly way to welcome a new colleague, @Arturo: bravo! – Georges Elencwajg Feb 13 '12 at 21:18
Why don't you try the quadratic formula on $y^2+hy-f=0$ and use the degree conditions to show that $h^2+4f$ is never a perfect square? – Parsa Feb 14 '12 at 1:42