The germ induced by an irreducible polynomial Let $P\in\mathbb{C}[z_1,z_2,\ldots,z_n]$ be an irreducible polynomial. Let $a\in\mathbb{C}^n$ be such that $P(a)=0.$ Consider the germ of holomorphic functions at the point $a,$ denoted by $\mathcal{O}_a.$ I have the following question: Is the germ induced by $P$ at $a$ irreducible in $\mathcal{O}_a$? Remark: If possible please suggest a reference to understand the problem.

No, consider $y^2-x^3-x^2 \in \mathbb{C}[x,y]$. It is an irreducible polynomial (by Eisenstein's criterion), but not an irreducible element of $\mathcal{O}_{(0,0)}$, since we have $y^2-x^3-x^2 = (y-x \sqrt{1+x})(y+x \sqrt{1+x})$. (Geometrically, the we look at the curve $V(y^2=x^3+x^2)$, and the completion reveals its two tangent directions at the origin.)