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  "body": "\n## Summary\n\nThe discriminant of an [[algebraic number field|MWBDFS2KJ5]] $K$ is a nonzero integer $d_K$ that measures the size of the ring of integers of $K$.\nView the ring of integers as a lattice in a real vector space.\nUp to sign and a power of $2$, $d_K$ is the square of the volume of a fundamental domain of this lattice.\nA prime number ramifies in $K$ exactly when it divides $d_K$.\nThe discriminant appears in the [[Minkowski bound|1TQ2F9CFVE]], the [[class number formula|55XX5A7XZV]] and the [[Dedekind zeta function|KXS5DQDFYD]].\n\n## Explanations\n\n### L1 human\n\nTake a basis of the ring of integers of $K$ and place it inside the complex numbers in all possible ways.\nThe discriminant is the squared determinant of the square table of numbers that this gives.\nFor the Gaussian integers $\\Z[i]$, with basis $1, i$, the table is $\\begin{pmatrix} 1 \u0026 i \\\\ 1 \u0026 -i \\end{pmatrix}$ and the discriminant is $(-2i)^2 = -4$.\nThe only prime dividing $-4$ is $2$, and $2 = -i(1+i)^2$ is the only prime that becomes a square in $\\Z[i]$, up to a unit.\nA large discriminant means that the field is arithmetically complicated.\n",
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