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If $f$ is an irreducible polynomial over $ K ⊆ ℂ $ and at least one zero of $f$ is expressible by radicals, prove that every zero of $f$ is expressible by radicals.

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My solution is:

$α$ is expressible by radicals means that there exists a sequence of intermediate fields $K ⊆ K(β_1)⊆ ...⊆ K(β_1,...,β_n)=K(α)$ such that $β^{m_j}_j ∈ K(β_1,..,β_{j-1})$.

But we can use a nice theorem here actually [ Suppose $ K(a):K$ and $Κ (b) : Κ$ are simple algebraic extensions, such that a and b have the same minimal polynomial $m$ over $K$. Then the two extensions are isomorphic, and the isomorphism of the large fields can be taken to map $a$ to $b$ (and to be the identity on $K$)].

Hence if $τ$ is the isomorphism between $ K(α)$ and $ K(α_1)$ and s.t $τ(α)= α_1$ then we have this sequence of intermediate fields $K ⊆ K(τ(β_1))⊆ ...⊆ K(τ(β_1),...,τ(β_n))=K(τ(α))= K(α_1)$ such that $τ(β_j^{m_j})=τ(β_j)^{m_j} ∈ K(τ(β_1),..,τ(β_{j-1}))$.

And we done!

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