Is this a prime Ideal? I wish to see wether $J=(uw -v^2, u^3 - vw, w^3 -u^5)\subset\mathbb{C}[u,v,w]$ is a prime ideal. Can somebody give me a hint to do this?
Edit: More generally, I wonder wether $V(J)$, the algebraic set given by the vanishing locus of $J$, is irreducible.
 A: $J$ is prime if and only if $\Bbb C[u,v,w]/J$ is an integral domain.
So, what does $\Bbb C[u,v,w]/J$ looks like ?
Since in the quotient, $w^3 = u^5$, $w$ should look like a $5$th power. 
This should lead you to define the map $\Bbb C[u,v,w] \to \Bbb C[x], P(u,v,w) \mapsto P(x^3,x^4,x^5)$.
Now, the tedious part is to prove that the kernel of this map is exactly $J$, so that this gives an isomorphism $\Bbb C[u,v,w]/J \to \Bbb C[x^3,x^4,x^5]$, which is obviously an integral domain.
A: Since this question has been tagged algebraic geometry, I should say a little bit about how we determine if the vanishing locus of your ideal is a variety. Now  a first step of course would be to attempt to show that $I$ is prime. If this were the case, the Nullstellensatz then gives that $\mathcal{I}\mathcal{V}(I) = \operatorname{rad} I = I$  so that $\mathcal{V}(I)$ is a variety. However it may be the case that $I$ is not prime, so there is an alternative way as follows.
In your case we wish to determine if $$V = \mathcal{V}(uw - v^2, u^3 - vw, w^3 - u^5)$$
is a variety. Then any point $(x,y,z) \in V$ necessarily satisfies
$$xz = y^2,\hspace{2mm} x^3 = yz,\hspace{2mm} z^3 = x^5.$$
We may suppose that $(x,y,z) \neq 0$ (for the origin is already in $V$) to get that $x^4 = y^3, z^3 = x^5$ and so 
$$x = \left( \frac{y}{x} \right)^3,\hspace{2mm} y = \left( \frac{y}{x} \right)^4, \hspace{2mm} z = \left( \frac{y}{x} \right)^5. $$
This means to say that $V$ is actually the image of the polynomial map
$$\begin{eqnarray*} \varphi : &\Bbb{A}^1& \longrightarrow V \\
&t& \mapsto (t^3,t^4,t^5). \end{eqnarray*}$$
Since $\Bbb{A}^1$ is a variety it follows that $V$ is a variety too. Notice how I don't need to invoke any Nullstellensatz to prove that $V$ was a variety this way!
Edit: To the person that downvoted my answer, why is it wrong? If you don't tell me what's wrong how am I supposed to improve?
