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  "body": "\n## Summary\n\nA radical extension is a field extension built by adjoining roots one at a time.\nAt each step one adjoins an $n$-th root of an element already present.\nEach of its elements can be written from elements of the base field with the four field operations and root extraction.\nA polynomial is [[solvable by radicals|J7YS0DA3HQ]] when its roots lie in a radical extension, and [[Galois' theorem|VHCWCFA3MX]] turns this into a property of a group.\n\n## Explanations\n\n### L1 human\n\nStart with the rational numbers and allow yourself to take square roots, cube roots and so on, one step at a time.\nEvery field you reach in this way is a radical extension of $\\Q$.\nFor example, $\\Q(\\sqrt{2})$ is radical, and so is $\\Q(\\sqrt{2}, \\sqrt[3]{1 + \\sqrt{2}})$, built in two steps.\nA number like $\\sqrt[3]{1 + \\sqrt{2}}$ is exactly the kind of expression that a formula for the roots of a polynomial produces.\n",
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