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 φ ∈ G and a ≤ b in R show that φ ( a ) ≤ φ ( b ). [Hint: b-a is a square in R .] (b) Show that G = 1. [Hint: If not choose φ ∈ G and a ∈ R such that φ ( a ) 6 = a . Choose b ∈ Q between a and φ ( a ).] (6) Let F ⊆ K be a ﬁeld extension. (a) Suppose K = F ( x ) is simple transcendental, and show that there are inﬁnitely many intermediate ﬁelds F ⊆ L ⊆ K . (b) Prove the same conclusion as (a) whenever [ K : F ] is inﬁnite....
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- Fall '08
- Algebra, Field extension, j G, Dr. Matthew Macauley, Matthew Macauley HW, simple transcendental extension