Monday, 5 September 2016

abstract algebra - Degree of field extensions in mathbbQ with two algebraic elements



I've got a question regarding the degree of field extensions.



The question is as follows: given two algebraic elements α,βQ, 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 KLM, 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]



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