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Homeradical extension

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# radical extension

A radical tower is a field extension $L/F$ which has a filtration

$F=L_{0}\subset L_{1}\subset\cdots\subset L_{n}=L$ |

where for each $i$, $0\leq i<n$, there exists an element $\alpha_{i}\in L_{{i+1}}$ and a natural number $n_{i}$ such that $L_{{i+1}}=L_{i}(\alpha_{i})$ and $\alpha_{i}^{{n_{i}}}\in L_{i}$.

A radical extension is a field extension $K/F$ for which there exists a radical tower $L/F$ with $L\supset K$. The notion of radical extension coincides with the informal concept of solving for the roots of a polynomial by radicals, in the sense that a polynomial over $K$ is solvable by radicals if and only if its splitting field is a radical extension of $F$.

Synonym:

radical tower

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Definition

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## Mathematics Subject Classification

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