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  "body": "\n## Definition\n\nA quadratic integer is a complex number $x$ that satisfies an equation $x^2 + bx + c = 0$ with integers $b$ and $c$.\nEquivalently, $x = \\dfrac{-b \\pm \\sqrt{b^2 - 4c}}{2}$ for some integers $b$ and $c$.\n\n## Remarks\n\nEach integer $n$ is a quadratic integer, since it is a root of $x^2 - 2nx + n^2 = 0$.\nA quadratic integer that is not an integer is an [[algebraic number|0MB4TGMXG6]] of degree $2$.\nIt lies in a quadratic field $\\mathbb{Q}(\\sqrt{D})$ with $D$ a squarefree integer.\nThe [[Gaussian integers|0B268GVX95]] are the quadratic integers in $\\mathbb{Q}(i)$. Another example is the golden ratio $\\frac{1 + \\sqrt{5}}{2}$, a root of $x^2 - x - 1 = 0$.\nFor $D \\equiv 2, 3 \\pmod 4$, the quadratic integers in $\\mathbb{Q}(\\sqrt{D})$ form the ring $\\mathbb{Z}[\\sqrt{D}] = \\{ a + b\\sqrt{D} : a, b \\in \\mathbb{Z} \\}$.\n\n## Explanations\n\n### L1 human\n\nA quadratic integer is a root of a quadratic equation with integer coefficients and leading coefficient $1$.\nNumbers such as $\\sqrt{2}$, $i$ and $1 + \\sqrt{3}$ are examples. They extend the integers in the same way that $\\sqrt{2}$ extends the rationals.\n",
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