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  "body": "\n## Statement\n\nLet $L/K$ be a finite [[Galois extension|0DS2F7GK05]] with [[Galois group|0XVR72M51M]] $G = \\Gal(L/K)$.\nThen there is an element $\\alpha \\in L$ such that the family $(\\sigma(\\alpha))_{\\sigma \\in G}$ is a [[basis|0VTQTPQGAV]] of $L$ as a $K$-vector space.\n\n## Remarks\n\nSuch a basis is called a normal basis, and $\\alpha$ is a normal basis generator.\nEquivalently, $L$ is isomorphic to the group algebra $K[G]$ as a module over $K[G]$.\n\nWhen $K$ is infinite, write $G = \\set{\\sigma_1, \\ldots, \\sigma_n}$ and fix a basis $b_1, \\ldots, b_n$ of $L$.\nThe family $(\\sigma_i(\\alpha))$ is a basis exactly when $\\det\\left(\\sigma_i^{-1}\\sigma_j(\\alpha)\\right) \\ne 0$; writing $\\alpha = \\sum x_k b_k$ makes this determinant a polynomial in $x_1, \\ldots, x_n$.\nAn argument in the spirit of [[Dedekind's lemma|N43G50YHG2]], using that $K$ is infinite, shows that this polynomial is not zero.\nA nonzero polynomial over an infinite field has a non-root in $K^n$.\nWhen $K$ is finite, $G$ is cyclic, generated by the Frobenius $\\varphi$.\nDedekind's lemma shows that the minimal polynomial of the $K$-linear map $\\varphi$ is $x^n - 1$, and the theory of cyclic modules then gives the normal basis.\n\nExample: $L = \\Q(i)$ over $K = \\Q$, with $G = \\set{\\id, \\sigma}$ and $\\sigma(i) = -i$.\nFor $\\alpha = 1 + i$, the family $1 + i, 1 - i$ is a basis of $\\Q(i)$ over $\\Q$.\nThe choices $\\alpha = 1$ and $\\alpha = i$ fail: they give $1, 1$ and $i, -i$, which are linearly dependent.\n\n## Explanations\n\n### L1 human\n\nA Galois extension has a group of symmetries.\nThe normal basis theorem says that one well-chosen element, together with all its images under the symmetries, already forms a basis.\nSo the whole field is spanned by the conjugates of a single element.\n\n### L2 human\n\nIn coordinates of a normal basis, the Galois group acts by permuting the coordinates, which makes many computations simple.\nOver finite fields, normal bases make the Frobenius map $x \\mapsto x^q$ a cyclic shift of coordinates, which is used in computer arithmetic.\nThe theorem says that $L$ is a free $K[G]$-module of rank $1$.\nWhether the ring of integers of a number field is free over $\\Z[G]$ is a much subtler question of the theory of Galois modules.\n",
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