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  "body": "\n## Statement\n\nLet $L/K$ be a [[finite|0NRYPM9Y27]] [[separable extension|0N019F6RE9]].\nThen $L/K$ is a [[simple extension|0P6SBR7YM2]]: there is $\\theta \\in L$ with $L = K(\\theta)$.\n\n## Remarks\n\nEvery finite extension of a field of characteristic $0$ is separable, so it is simple. For example, $\\mathbb{Q}(\\sqrt{2}, \\sqrt{3}) = \\mathbb{Q}(\\sqrt{2} + \\sqrt{3})$.\nFor an infinite field $K$, the proof shows that $\\theta = a + c b$ works for all but finitely many $c \\in K$.\nFor finite fields, a [[primitive element|0YAA9CJQXF]] of $L$ generates $L$ over $K$.\nWithout separability the statement can fail: there are finite extensions of degree $p^2$ in characteristic $p$ with no primitive element.\n\n## History\n\nÉvariste Galois sketched a proof for splitting fields over the rationals in his First Memoir of 1831, published in 1846.\nErnst Steinitz proved the theorem in its modern form in 1910.\nEmil Artin later developed Galois theory without primitive elements.\n\n## Explanations\n\n### L1 human\n\nA finite separable extension, even one built from many elements, can be generated by one element.\nSo many statements about finite extensions reduce to the case of one algebraic element and its minimal polynomial.\n",
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