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  "body": "\n## Definition\n\nLet $K$ be a [[field|0QA8CHH8MG]] and let $f \\in K[x]$ be [[irreducible|00X2A9G9W3]].\nThe field obtained by adjoining a root of $f$ to $K$ is the [[quotient ring|0VFW4Y69BT]] $K[x]/(f)$.\nThe map $K \\to K[x]/(f)$, $c \\mapsto c + (f)$, identifies $K$ with a subfield, and the class $\\alpha = x + (f)$ is a root of $f$.\n\n## Remarks\n\nThe quotient is a field, because [[$f$ is irreducible|0AWMPX35CW]].\nIts degree over $K$ is $\\deg f$, with the basis $1, \\alpha, \\dots, \\alpha^{\\deg f - 1}$.\nFor example, adjoining a root of $x^2 + 1$ to $\\mathbb{R}$ gives a field isomorphic to $\\mathbb{C}$.\n\n## Explanations\n\n### L1 human\n\nYou can create a root of a polynomial that has none.\nCompute with polynomials in $x$, and replace $f(x)$ by zero each time it appears.\nThis is how $\\mathbb{C}$ arises from $\\mathbb{R}$ and $x^2 + 1$.\n",
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