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Math Help - A simple problem in algebra from feilds

  1. #1
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    A simple problem in algebra from feilds

    I need a rigourous proof for the following problem if possible:-
    Let L be a finite extension of F and let K be a subfeild of L such that it contains F.
    Show that [K:F] | [L:F] .

    Please help.
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  2. #2
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    Quote Originally Posted by upanshu View Post
    I need a rigourous proof for the following problem if possible:-
    Let L be a finite extension of F and let K be a subfeild of L such that it contains F.
    Show that [K:F] | [L:F] .

    Please help.
    If F\subseteq K\subseteq L and K/L is finite then L/K and K/F is finite. Furthermore, [L:F] = [L:K][K:F]. Therefore, [K:L] divides [L:F]. To see this, note if \{ a_1,...,a_n\} is basis for K/F and \{b_1,...,b_m\} is basis for L/K then \{ a_ib_j\} is a basis for L/F.
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