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  "body": "\n## Definition\n\nLet $L/K$ be a [[field extension|0WQ1A8SHTZ]].\nThe field $L$ is a vector space over $K$, with the addition of $L$ and the multiplication $K \\times L \\to L$ of $L$.\nThe degree of $L/K$ is the dimension of this vector space, and it is written $[L : K]$.\n\n## Remarks\n\nThe degree is a positive integer or an infinite cardinal.\nOne has $[L : K] = 1$ if and only if $L = K$.\nExamples: $[\\mathbb{C} : \\mathbb{R}] = 2$, with the basis $1, i$. Also, $[\\mathbb{Q}(\\sqrt{2}) : \\mathbb{Q}] = 2$, with the basis $1, \\sqrt{2}$.\nThe field $\\mathbb{R}$ has infinite degree over $\\mathbb{Q}$, because $\\mathbb{Q}$ is countable and $\\mathbb{R}$ is not.\n\n## Explanations\n\n### L1 human\n\nThe degree counts how many elements of the big field you need, as a basis, to write every element with coefficients in the small field.\nEvery complex number is $a + b i$ with $a, b$ real, so the degree of $\\mathbb{C}$ over $\\mathbb{R}$ is $2$.\n",
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