Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
add comment

1 Answer

up vote 1 down vote accepted

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!

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.