Let $A$ be an irreducible analytic hypersurface of $\mathbb{C}^n$ contains the origin $0$. Then there is a neighborhood $U$ of $0$ and a holomorphic function $f$ on $U$ such that $A\cap U=\{x\in U \, |\, f(x)=0\}$ and the germ of $f$ in $\mathcal{O}_{\mathbb{C}^n,0}$ generates the ideal of all stalks that annihilates $A$. The germ $f_0$ is always square-free but in general not irreducible.
My question is that doses every irreducible element in $\mathcal{O}_{\mathbb{C}^n,0}$ can be gained from an irreducible analytic hypersurface of $\mathbb{C}^n$ in this way? Note that this is obviously true for $n=1$.
All comments are welcome. Thanks a lot.
Now I know the following fact: if the germ of $f$ at $0$ is irreducible then the vanishing set of $f$ in some open neighborhood $\Omega$ of $0$ is irreducible as a analytic subset of $\Omega$.
From this, one can deduce that every irreducible element of $\mathcal{O}_{\mathbb{C}^n,0}$ can be gained from an irreducible analytic hypersurface of some open neighborhood $\Omega$ of $0$.