An ordinary node has two smooth branches

Apply [F1] to $F(x,w)=w^2-(1+x)$ at the point $(x_0,w_0)=(0,1)$: $F(0,1)=0$ and ${\mathrm{\partial }}_{w}F(0,1)=2\mathrm{\ne }0$, so there is a holomorphic function $s$ on a neighbourhood of $0$ with $s(0)=1$ and $s(x{)}^2=1+x$ identically. In particular $s$ is a unit of ${\mathcal{O}}_{\mathbb{C},0}$ and $s(x)\mathrm{\ne }0$ for $\mathrm{\mid }x\mathrm{\mid }$ small.

Put $g_{\pm }:=y\mp xs(x)$, the two germs determined by the unit $s$ of step 1.1; each is a monic polynomial of degree $1$ in $y$ with coefficient $\mp xs(x)$ in ${\mathcal{O}}_{\mathbb{C},0}$ vanishing at $x=0$. Each $g_{\pm }$ is irreducible in ${\mathcal{O}}_{{\mathbb{C}}^2,0}$. Suppose $g_{+}=gh$ with $g,h$ nonunits. Since $g_{+}$ is regular in $y$ of order $1$, [F3] makes $g$ and $h$ regular with Weierstrass polynomials $G,H$ of positive degrees satisfying $W_{g_{+}}=GH$; but $W_{g_{+}}=g_{+}$ has degree $1$, while $\mathrm{deg}G+\mathrm{deg}H\ge 2$, a contradiction. The same argument applies to $g_{-}$.

With $s$ as in step 1.1, $f=y^2-x^{2}s(x{)}^2=(y-xs(x))(y+xs(x))$. The two factors $g_{\pm }:=y\mp xs(x)$ are Weierstrass polynomials of degree $1$ in $y$: each is monic of degree $1$ with its coefficient $\mp xs(x)$ lying in ${\mathcal{O}}_{\mathbb{C},0}$ and vanishing at $x=0$. The germ $f$ is itself a Weierstrass polynomial of degree $2$: it is monic of degree $2$ in $y$, its coefficients $-x^2(1+x)$ vanish at the origin, and $f(0,y)=y^2$, so it is regular in $y$ of order $2$ and is its own Weierstrass preparation by [F2].

The zero germs of $g_{+}$ and $g_{-}$ are distinct: $Z(g_{+})$ is the graph of ${\phi }_{+}(x)=xs(x)$ and $Z(g_{-})$ is the graph of ${\phi }_{-}(x)=-xs(x)$ over the $x$-coordinate. Since $s(x)\mathrm{\ne }0$ for $\mathrm{\mid }x\mathrm{\mid }$ small by step 1.1, the two graphs intersect exactly where $xs(x)=0$, that is, only at $x=0$, and for every small $x\mathrm{\ne }0$ the values $xs(x)$ and $-xs(x)$ differ; hence the two set germs at the origin are distinct, and neither is contained in the other.

The germ $f$ is reduced, and $X=Z(f)$ has exactly the two irreducible components $Z(g_{+})$ and $Z(g_{-})$. Indeed $f=g_{+}{g}_{-}$ exhibits $f$ as a product of two nonassociate irreducibles, each occurring once; by the uniqueness of factorisation in the UFD [F6], no irreducible germ divides $f$ twice, so $f$ is reduced, and the factorisation of a reduced germ into pairwise nonassociate irreducibles has all exponents one and is unique up to order and associates. Applying the decomposition statement [F4] to $f=g_{+}{g}_{-}$ gives $X=Z(g_{+})\cup Z(g_{-})$ with $Z(g_{+})$ and $Z(g_{-})$ the two irreducible components of $X$.

Each branch is a holomorphic hypersurface graph and carries an injective convergent parametrisation. For $\mathrm{\mid }t\mathrm{\mid }$ small, ${\gamma }_{+}(t)=(t,ts(t))$ is holomorphic with ${\gamma }_{+}(\mathrm{\Delta })=Z(g_{+})\cap \{\mathrm{\mid }x\mathrm{\mid }<\rho \}$ for a suitable radius $\rho >0$, since $y-xs(x)=0$ is exactly the graph $y=xs(x)$; similarly ${\gamma }_{-}(t)=(t,-ts(t))$ has image $Z(g_{-})\cap \{\mathrm{\mid }x\mathrm{\mid }<\rho \}$. Both maps are injective because their first coordinate is $t$, and both are restrictions of the holomorphic function $s$ of step 1.1, hence convergent. Every point of each branch is therefore regular as a point of that branch by [F5]. At a point other than the origin, step 2.3 separates the two graphs, so $X$ locally equals the branch through that point and is regular there. This does not assert regularity of their union at the origin.

The origin is the only singular point of $X$ near $0$. By step 3.1 the reduced defining germ of $X$ is $f$, and

Steps 2.1 to 4.1 establish the assertions: $y^2-x^2(1+x)$ is a reduced equation of $X$ whose irreducible components are the two smooth branches $y=\pm x\sqrt{1+x}$ described by the unit square root $s$ of $1+x$; the origin is their only intersection and the only singular point of the germ, with distinct tangent directions, so $X$ is an ordinary node; and each branch carries the convergent parametrisation $t\mapsto (t,\pm ts(t))$ with first coordinate $t$.