Definition 15.115.1. Let $A \to B$ be an extension of discrete valuation rings with fraction fields $K \subset L$.

1. We say a finite field extension $K_1/K$ is a weak solution for $A \subset B$ if all the extensions $(A_1)_{\mathfrak m_ i} \subset (B_1)_{\mathfrak m_{ij}}$ of Remark 15.114.1 are weakly unramified.

2. We say a finite field extension $K_1/K$ is a solution for $A \subset B$ if each extension $(A_1)_{\mathfrak m_ i} \subset (B_1)_{\mathfrak m_{ij}}$ of Remark 15.114.1 is formally smooth in the $\mathfrak m_{ij}$-adic topology.

We say a solution $K_1/K$ is a separable solution if $K_1/K$ is separable.

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