I'm trying to prove that the affine curve $X\subset\mathbb{A}^3$ given by $\alpha:\mathbb{A}^1\to\mathbb{A}^3$, $t\mapsto(t^3,t^4,t^5)$, is not isomorphic to a plane curve.
Here is what I've done: it is enough to show that the dimension of $\mathcal{O}_{X,(0,0,0)}$ is greater than 2. As $\mathcal{O}_{X,(0,0,0)}=k[X]_{\mathfrak{m_0}}$, where $k[X]$ is the coordinate ring $k[x,y,z]/(x^4-y^3,x^5-z^3)$ and $\mathfrak{m_0}=\{f\in k[X]:f(0,0,0)=0\}=(\bar x,\bar y,\bar z)$, and as the prime ideals in $k[X]_{\mathfrak{m_0}}$ are in bijective correspondence with the prime ideals in $k[X]$ which are contained in $\mathfrak{m_0}$, it will suffice to find a chain of length 4 of prime ideals between $(0)$ and $\mathfrak{m_0}$ in $k[X]$. I've thought of $(0)\subset (\bar x)\subset (\bar x,\bar y)\subset \mathfrak{m_0}$.
Could somebody point out some possible flaw? I've never got along with irreducibility issues so my apologizes for every stupid confussion. Thanks in advance!