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  "body": "\n## Statement\n\nLet $L/K$ be a [[field extension|0WQ1A8SHTZ]] and let $a \\in L$ be algebraic over $K$ with [[minimal polynomial|0GK1WB9R98]] $m_a$ of degree $n$.\nThen $[K(a) : K] = n$, and $1, a, a^2, \\dots, a^{n-1}$ is a basis of $K(a)$ over $K$.\n\n## Remarks\n\nThe proof: by [[the structure theorem|0ZN39QDTP9]], $K(a) \\cong K[x]/(m_a)$. By polynomial division, the classes of $1, x, \\dots, x^{n-1}$ are a basis of $K[x]/(m_a)$.\nFor example, $[\\mathbb{Q}(\\sqrt[3]{2}) : \\mathbb{Q}] = 3$, with the basis $1, \\sqrt[3]{2}, \\sqrt[3]{4}$.\nWith the [[tower law|0A7KJ68XHT]], the degree of every element of a finite extension $L/K$ divides $[L : K]$.\n\n## Explanations\n\n### L1 human\n\nThe degree of the field you get by adding one algebraic element is the degree of its minimal polynomial.\nEach element of $K(a)$ is a polynomial in $a$ of degree less than $n$, in exactly one way.\n",
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