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  "body": "\n## Definition\n\nLet $L/K$ be a [[finite extension|0NRYPM9Y27]] of degree $n$ and let $b_1, \\ldots, b_n \\in L$.\nLet $\\mathrm{Tr}_{L/K}$ be the [[field trace|73XV4RP5PT]].\nThe *discriminant* of the family $(b_1, \\ldots, b_n)$ is\n$$d_{L/K}(b_1, \\ldots, b_n) = \\det\\left(\\mathrm{Tr}_{L/K}(b_i b_j)\\right)_{1 \\le i, j \\le n} \\in K,$$\nwhere $\\det$ is the [[determinant|0C3CFHDN6R]].\nThe *discriminant of $L/K$* is the class of $d_{L/K}(b_1, \\ldots, b_n)$ modulo $(K^\\times)^2$, for any [[basis|0VTQTPQGAV]] $b_1, \\ldots, b_n$ of $L$ over $K$.\n\n## Remarks\n\nIf $c_1, \\ldots, c_n$ is another basis and $c = Pb$ with $P \\in GL_n(K)$, then $d(c) = \\det(P)^2\\, d(b)$, so the class modulo squares is well defined.\nIf $L/K$ is separable with embeddings $\\sigma_1, \\ldots, \\sigma_n$ into an algebraic closure, then $d(b) = \\det\\left(\\sigma_i(b_j)\\right)^2$.\nThe discriminant of a basis is nonzero if and only if $L/K$ is separable.\nIf $L = K(\\alpha)$ and $f$ is the minimal polynomial of $\\alpha$, then $d(1, \\alpha, \\ldots, \\alpha^{n-1})$ is the [[discriminant|0CCPN09EMJ]] of $f$.\nFor a number field $L$ over $\\Q$, the discriminant of a $\\Z$-basis of the ring of integers is an integer independent of the basis, the discriminant of the number field.\n\nExample: $L = \\Q(i)$, $K = \\Q$, basis $1, i$.\nCheck: $\\mathrm{Tr}(1) = 2$, $\\mathrm{Tr}(i) = 0$ and $\\mathrm{Tr}(i^2) = -2$, so $d(1, i) = \\det\\begin{pmatrix} 2 \u0026 0 \\\\ 0 \u0026 -2 \\end{pmatrix} = -4$, the discriminant of $x^2 + 1$.\n\nNon-example: for $K = \\F_p(t)$ and $L = K(s)$ with $s^p = t$, every trace is $0$, so the discriminant of every basis is $0$; the extension is not separable.\n",
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