When I read the definition of a radical extension, I thought: What is the difference with a algebraic extension?
Could you give me an example of an algebraic extension which is not a radical extension?
Thanks.
2025-01-12 23:54:57.1736726097
Example of an algebraic extension which is not a radical extension
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The only way for a cubic extension of the rationals to be radical is for it to be a pure cubic extension, that is, it has to be ${\bf Q}(r^{1/3})$ for some rational $r$ (indeed, we may take $r$ to be an integer). But most cubic extensions are not pure cubics. Maybe the simplest example would be a cyclic cubic field, such as the one generated over the rationals by $\cos(2\pi/7)$ or $\cos(2\pi/9)$. These can't be pure cubic, since they are normal extensions, which pure cubics aren't.
Let $K$ be the splitting field of $f(x)=x^5-x-1$ over the rationals. It can be proved that the zeros of $f$ can't be expressed in radicals at all. That means $K$ isn't even contained in a radical extension.