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  "body": "\n## Definition\n\nLet $K$ be a [[field|04C5ZBG1WT]] that contains the field $\\Q$ of [[rational numbers|0NCJ5WXN6Z]].\nThen $K$ is an *algebraic number field* (or *number field*) if it has finite dimension as a vector space over $\\Q$.\nThis dimension is the [[degree|0EAJ2XXADC]] $[K : \\Q]$ of the [[field extension|0EQWVHF3W5]] $K / \\Q$, also called the degree of $K$.\n\n## Remarks\n\nExample: $K = \\Q(i) = \\set{a + bi : a, b \\in \\Q}$, a subfield of $\\C$.\nIt contains $\\Q$, and $1, i$ is a basis over $\\Q$, so $[K : \\Q] = 2$ and $K$ is a number field.\nNon-example: $\\R$ contains $\\Q$, but it is not of finite dimension over $\\Q$, since a finite-dimensional space over $\\Q$ is countable and $\\R$ is not.\nAnother non-example: the field of all algebraic numbers is an algebraic extension of $\\Q$ of infinite degree.\nEvery element of a number field is an [[algebraic number|0MNF0BYFQC]], and by the primitive element theorem every number field is $\\Q(\\alpha)$ for one algebraic number $\\alpha$.\nThe elements of $K$ that are roots of a [[monic polynomial|0C3QBCR46M]] with integer coefficients form a subring $\\mathcal{O}_K$, the ring of integers of $K$.\n",
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