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  "body": "\n## Summary\n\nThe separable degree $[L:K]_s$ of an algebraic extension $L/K$ counts the embeddings of $L$ into an [[algebraically closed field|0PF6TM8TWT]].\nOnly the embeddings that extend a fixed embedding of $K$ are counted.\nFor a finite extension it divides the [[degree|0EAJ2XXADC]] $[L:K]$, and the quotient is a power of the characteristic exponent, called the inseparable degree.\nIt equals $[L:K]$ exactly when $L/K$ is [[separable|0WCX7NB4N7]], and it equals the degree of the [[separable closure|JJ91A1V8NQ]] of $K$ in $L$.\nIt measures how much of an extension is visible to Galois theory.\n\n## Explanations\n\n### L1 human\n\nThe degree of an extension counts dimensions.\nThe separable degree counts the distinct ways to place the extension inside a big field such as $\\C$.\nFor $\\Q(\\sqrt{2})$ there are two ways, sending $\\sqrt{2}$ to $\\sqrt{2}$ or to $-\\sqrt{2}$, and the degree is also $2$.\nIn characteristic $p$, the extension of $\\F_p(t)$ by a $p$-th root of $t$ has degree $p$ but only one placement, because $x^p - t$ has only one root.\nSo its separable degree is $1$.\n",
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