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Unformatted text preview: S K and K is algebraic over F ( S ) show that there is a transcendence basis B for K over F with B S (5) (a) Let G = Gal( R / Q ). If G and a b in R show that ( a ) ( b ). [Hint: ba is a square in R .] (b) Show that G = 1. [Hint: If not choose G and a R such that ( a ) 6 = a . Choose b Q between a and ( a ).] (6) Let F K be a eld extension. (a) Suppose K = F ( x ) is simple transcendental, and show that there are innitely many intermediate elds F L K . (b) Prove the same conclusion as (a) whenever [ K : F ] is innite....
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 Fall '08
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 Algebra

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