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  "body": "\n## Statement\n\nLet $L/K$ be a [[field extension|0WQ1A8SHTZ]] and let $a \\in L$ be algebraic over $K$.\nThen the [[minimal polynomial|0GK1WB9R98]] $m_a$ of $a$ over $K$ is [[irreducible|00X2A9G9W3]] over $K$.\n\n## Remarks\n\nThe proof: if $m_a = g h$ with $g, h \\in K[x]$ of smaller degree, then $g(a) h(a) = 0$ in the field $L$. So $g(a) = 0$ or $h(a) = 0$, and this contradicts the minimality of $\\deg m_a$.\nConversely, a monic irreducible polynomial $f \\in K[x]$ with $f(a) = 0$ is the minimal polynomial of $a$.\nFor example, $x^3 - 2$ is irreducible over $\\mathbb{Q}$, so it is the minimal polynomial of $\\sqrt[3]{2}$ over $\\mathbb{Q}$.\n\n## Explanations\n\n### L1 human\n\nThe minimal polynomial cannot be factored over the base field.\nIf it could, one of the factors would already vanish at the element, and that factor would be a smaller polynomial.\n",
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