I've got a question regarding the degree of field extensions.
The question is as follows: given two algebraic elements α,β∈Q, assume that [Q(α):Q] and [Q(β):Q] are relatively prime. Show that [Q(α,β):Q]=[Q(α):Q]⋅[Q(β):Q].
We know that both [Q(α):Q] and [Q(β):Q] are finite because α and β are algebraic, so we could say [Q(α):Q]=n and [Q(β):Q]=m. We also know that [Q(α,β):Q] is finite. I'm assuming I have to use that given a field K and two field extensions L,M where K⊂L⊂M, we have [M:K]=[M:L]⋅[L:K]. But I'm not sure what to do next, because I don't know how to show that these extensions match these criteria.
Answer
Hint:
[Q(α,β):Q] is a common multiple of [Q(α):Q] and [Q(β):Q]
[Q(α,β):Q(α)]≤[Q(β):Q]
No comments:
Post a Comment