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  "body": "\n## Definition\n\nLet $L/K$ be a [[field extension|0WQ1A8SHTZ]] and let $a \\in L$ be [[algebraic|0NPW1DWECM]] over $K$.\nThe minimal polynomial of $a$ over $K$ is the [[monic polynomial|0BDX4MBKT4]] $m_a \\in K[x]$ of smallest [[degree|0DPF6WRZW6]] such that $m_a(a) = 0$.\n\n## Remarks\n\nThe minimal polynomial exists and is unique. If two monic polynomials of the smallest degree vanish at $a$, their difference has smaller degree and vanishes at $a$, so it is zero.\nThe minimal polynomial [[is irreducible|0NFEG48RDR]], and it [[divides each polynomial that vanishes at $a$|06K7SARZKR]].\nThe minimal polynomial of $\\sqrt{2}$ over $\\mathbb{Q}$ is $x^2 - 2$. Over $\\mathbb{R}$, it is $x - \\sqrt{2}$.\nThe minimal polynomial of $i$ over $\\mathbb{R}$ is $x^2 + 1$.\n\n## Explanations\n\n### L1 human\n\nAmong all the polynomial equations that an algebraic element satisfies, the minimal polynomial is the simplest one.\nIt depends on the base field: a bigger base field can give a smaller minimal polynomial.\n",
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