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radical extension (Definition)

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

$\displaystyle 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$.



"radical extension" is owned by djao.
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Other names:  radical tower

Attachments:
casus irreducibilis (Theorem) by pahio
expressible (Definition) by Wkbj79
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Cross-references: splitting field, solvable by radicals, radicals, polynomial, roots, natural number, filtration, field extension
There are 7 references to this entry.

This is version 3 of radical extension, born on 2002-01-05, modified 2004-01-09.
Object id is 1329, canonical name is RadicalExtension.
Accessed 4633 times total.

Classification:
AMS MSC: 12F10 (Field theory and polynomials :: Field extensions :: Separable extensions, Galois theory)
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