Remark 53.19.9 (Trivial quadratic extension). Let $k$ be a field. Let $(A, \mathfrak m, \kappa )$ be as in Remark 53.19.8 and let $\kappa '/\kappa$ be the associated separable algebra of degree $2$. Then the following are equivalent

1. $\kappa ' \cong \kappa \times \kappa$ as $\kappa$-algebra,

2. the form $q$ of Lemma 53.19.3 can be chosen to be $xy$,

3. $A$ has two branches,

4. the extension $A'/A$ of Lemma 53.19.4 has two maximal ideals, and

5. $A^\wedge \cong \kappa [[x, y]]/(xy)$ as a $k$-algebra.

The equivalence between these conditions has been shown in the proof of Lemma 53.19.4. If $X$ is a locally Noetherian $k$-scheme and $x \in X$ is a point such that $\mathcal{O}_{X, x} = A$, then this means exactly that there are two points $x_1, x_2$ of the normalization $X^\nu$ lying over $x$ and that $\kappa (x) = \kappa (x_1) = \kappa (x_2)$.

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