Are elliptic curves algebraic varieties? I got a short question.

Are elliptic cubes also algebraic varietes?

Say we have $E:y^2=x^3+5x=:f(x)$

Then we can $f(x)=x(x^2+5)$

So it can't be an algebraic variety..

I feel like I am totally misunderstanding the definition of algebraic varieties.

Can someone help? Thanks
Elliptic curves are very famous examples of algebraic varieties. You seem to have forgotten that the function can have more than one variable; take the function $f(x,y)=y^2-x(x^2+5)$, then $E$ is just the variety $V(f)$.
You are messing up the notation and confusing yourself. You could write the polynomial
$$
f(x) = x^3 + 5x \in \mathbb{Q}[x] \text{ or } \mathbb{C}[x]
$$
or you could write the polynomial
$$
f(x,y) = y^2 - x^3 - 5x \in \mathbb{Q}[x,y]
$$
Once you factor $f(x)$ you find that it equals $x(x^2 + 5) = x(x+i\sqrt{5})(x-i\sqrt{5})$. This should give you the hint that it defines a single point in $\mathbb{Q}$ while giving you three distinct points in $\mathbb{C}$. The other polynomial does define an ellitpic curve as a variety over $\mathbb{Q}$.



Also, you could ramp up this discussion and frame it into a scheme theoretic language. Then, you should consider the scheme
$$
X = \text{Spec}(\mathbb{Z}[x,y]/(y^2 - x^3 - 5x))
$$
If you write this as the functor $X = \text{Hom}_{\textbf{CRing}}(\mathbb{Z}[x,y]/(y^2 - x^3 - 5x),-)$, then
\begin{align*}
X(\mathbb{F}_q) &amp;= \{ (a,b) \in \mathbb{F}_q^2: b^2 - a^3-5a =0 \} \\
X(\mathbb{Q}_p) &amp;= \{ (a,b) \in \mathbb{Q}_p^2: b^2 - a^3-5a =0 \} \\
X(\mathbb{Q}) &amp;= \{ (a,b) \in \mathbb{Q}^2: b^2 - a^3-5a =0 \} \\
X(\mathbb{C}) &amp;= \{ (a,b) \in \mathbb{C}^2: b^2 - a^3-5a =0 \} 
\end{align*}
and so on...