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  "body": "\n## Summary\n\nAn algebraic number field, or number field, is a [[field extension|0EQWVHF3W5]] of the rational numbers of finite degree.\nNumber fields are the main objects of algebraic number theory.\nEach number field $K$ has a ring of integers $\\mathcal{O}_K$, a [[Dedekind domain|7NPY2CN5KB]] that plays the role of $\\Z$ inside $\\Q$.\nThe failure of unique factorisation in $\\mathcal{O}_K$ is measured by the [[ideal class group|WQ46BD3ARY]].\n\n## Explanations\n\n### L1 human\n\nStart with the fractions $\\Q$ and adjoin a root of a polynomial, for example $i = \\sqrt{-1}$ or $\\sqrt{2}$.\nThe numbers you can then form with $+$, $-$, $\\times$ and $\\div$ make up a number field.\nFor example, $\\Q(i)$ consists of the numbers $a + bi$ with $a, b$ rational.\nIt is a two-dimensional space over $\\Q$, with basis $1, i$.\nNumber theory in such fields explains facts about ordinary integers, such as which primes are sums of two squares.\n",
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