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  "body": "\n## Definition\n\nA [[field extension|0WQ1A8SHTZ]] $L/K$ is finite if its [[degree|0GS74FGFEK]] $[L : K]$ is finite.\nOtherwise $L/K$ is an infinite extension.\n\n## Remarks\n\nA finite extension is not the same as an extension of finite fields. For example, $\\mathbb{C}/\\mathbb{R}$ is finite, and both fields are infinite.\nThe extension $\\mathbb{Q}(\\sqrt[3]{2})/\\mathbb{Q}$ is finite of degree $3$.\nThe field $K(x)$ of rational functions is an infinite extension of $K$, because $1, x, x^2, \\dots$ are linearly independent over $K$.\n[[Finite extensions are algebraic|0DFYTNZ6AM]].\n\n## Explanations\n\n### L1 human\n\nA finite extension is one that is a finite-dimensional vector space over the smaller field.\nEach element of it is a combination of finitely many fixed elements, with coefficients in the smaller field.\n",
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