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  "body": "\n## Definition\n\nA *quadratic field* is an [[algebraic number field|MWBDFS2KJ5]] $K$ with [[degree|0EAJ2XXADC]] $[K : \\Q] = 2$.\n\n## Remarks\n\nEvery quadratic field is $\\Q(\\sqrt{d}) = \\set{a + b\\sqrt{d} : a, b \\in \\Q}$ for exactly one [[square-free integer|08X0VCR2NW]] $d$ with $d \\ne 0, 1$.\nIt is called real if $d \u003e 0$ and imaginary if $d \u003c 0$.\nExample: $\\Q(\\sqrt{2})$; the element $\\sqrt{2}$ is a root of the irreducible polynomial $x^2 - 2$, so $1, \\sqrt{2}$ is a basis and the degree is $2$.\nNon-example: $\\Q(\\sqrt[3]{2})$ has degree $3$, because $x^3 - 2$ is irreducible over $\\Q$; so it is a number field but not a quadratic field.\nThe ring of integers of $\\Q(\\sqrt{d})$ is $\\Z[(1 + \\sqrt{d})/2]$ if $d \\equiv 1 \\pmod 4$, and $\\Z[\\sqrt{d}]$ if $d \\equiv 2$ or $3 \\pmod 4$.\nThe discriminant is $d$ in the first case and $4d$ in the second.\n",
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