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Question Number 214098 by issac last updated on 28/Nov/24

Let F be Field of characteristic 0 L_i (i=1,2) be two algebraic extension of F , and L_1 L_2 be a field in F^� (where F^� is the algebraic closure of F) defined by {l_1 l_2 ∣l_i ∈L_i (i=1,2)} 1. show that if L_1 and L_2 are galois over F then L_1 L_2 is also Galois over F 2. show that if G(L_1 /F^ ) and G(L_2 /F^ ) are Solvable , then Gal(L_1 L_2 /F^ ) is also Solvable

$$\mathrm{Let}\:{F}\:\mathrm{be}\:\:\mathrm{Field}\:\mathrm{of}\:\mathrm{characteristic}\:\mathrm{0} \\ $$$${L}_{{i}} \:\left({i}=\mathrm{1},\mathrm{2}\right)\:\mathrm{be}\:\mathrm{two}\:\mathrm{algebraic}\:\mathrm{extension} \\ $$$$\mathrm{of}\:{F}\:,\:\mathrm{and}\:{L}_{\mathrm{1}} {L}_{\mathrm{2}} \:\mathrm{be}\:\mathrm{a}\:\mathrm{field}\:\mathrm{in}\:\bar {{F}}\: \\ $$$$\left(\mathrm{where}\:\bar {{F}}\:\:\mathrm{is}\:\mathrm{the}\:\mathrm{algebraic}\:\mathrm{closure}\:\:\mathrm{of}\:{F}\right) \\ $$$$\mathrm{defined}\:\mathrm{by}\:\left\{{l}_{\mathrm{1}} {l}_{\mathrm{2}} \mid{l}_{{i}} \in{L}_{{i}} \:\left({i}=\mathrm{1},\mathrm{2}\right)\right\} \\ $$$$\mathrm{1}.\:\mathrm{show}\:\mathrm{that}\:\mathrm{if}\:{L}_{\mathrm{1}} \:\mathrm{and}\:{L}_{\mathrm{2}} \:\mathrm{are}\:\mathrm{galois}\:\mathrm{over}\:{F} \\ $$$$\mathrm{then}\:{L}_{\mathrm{1}} {L}_{\mathrm{2}} \:\mathrm{is}\:\mathrm{also}\:\mathrm{Galois}\:\mathrm{over}\:{F} \\ $$$$\mathrm{2}.\:\mathrm{show}\:\mathrm{that}\:\mathrm{if}\:{G}\left({L}_{\mathrm{1}} /{F}^{\:} \right)\:\mathrm{and}\:{G}\left({L}_{\mathrm{2}} /{F}^{\:} \right) \\ $$$$\mathrm{are}\:\mathrm{Solvable}\:,\:\mathrm{then}\:\mathrm{Gal}\left({L}_{\mathrm{1}} {L}_{\mathrm{2}} /{F}^{\:} \right)\:\mathrm{is}\:\mathrm{also} \\ $$$$\mathrm{Solvable} \\ $$

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