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  "body": "\n## Summary\n\nA quadratic field is an [[algebraic number field|MWBDFS2KJ5]] of degree $2$ over $\\Q$.\nEach one is $\\Q(\\sqrt{d})$ for a unique square-free integer $d \\ne 0, 1$.\nQuadratic fields are the simplest number fields beyond $\\Q$, and their arithmetic is closely tied to [[binary quadratic forms|SBDSYA037Q]] and to [[Pell's equation|0AF3DANJ76]].\n\n## Explanations\n\n### L1 human\n\nTake the rational numbers and add one square root, such as $\\sqrt{2}$ or $\\sqrt{-5}$.\nThe new numbers have the form $a + b\\sqrt{d}$ with $a, b$ rational, and they can be added, multiplied and divided like ordinary fractions.\nWhen $d \u003e 0$ the field sits inside the real numbers and is called real; when $d \u003c 0$ it is called imaginary.\nAlready here unique factorisation can fail: in $\\Q(\\sqrt{-5})$ one has $6 = 2 \\cdot 3 = (1 + \\sqrt{-5})(1 - \\sqrt{-5})$.\n",
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