Definition 15.46.1. Let $p$ be a prime number. Let $k \to K$ be an extension of fields of characteristic $p$. Denote $kK^ p$ the compositum of $k$ and $K^ p$ in $K$.

1. A subset $\{ x_ i\} \subset K$ is called p-independent over $k$ if the elements $x^ E = \prod x_ i^{e_ i}$ where $0 \leq e_ i < p$ are linearly independent over $kK^ p$.

2. A subset $\{ x_ i\}$ of $K$ is called a p-basis of $K$ over $k$ if the elements $x^ E$ form a basis of $K$ over $kK^ p$.

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