f09_mthsc852_hw16

# f09_mthsc852_hw16 - S ⊆ K and K is algebraic over F S...

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MTHSC 851/852 (Abstract Algebra) Dr. Matthew Macauley HW 16 Due Friday, November 1st, 2009 (1) Suppose F E K , F L K , Gal = G ( K/F ), J G , and H G (a) Show that G ( E L ) = G E G L and F ( J H ) = F J F H . (b) Show that [ E L : F ] [ E : F ][ L : F ]. (2) A ﬁeld F is called perfect if either char F = 0 or else char F = p and F = F p = { a p : a F } . (a) If F is ﬁnite show that the map a 7→ a p is a monomorphism and conclude that F is perfect. (b) Show that the ﬁeld Z p ( t ) of rational functions in the indeterminate t is not perfect. (c) Show that a ﬁeld F is perfect if and only if every ﬁnite extension K of F is separable over F , and hence every f ( x ) F [ x ] is separable. (3) Let F be any inﬁnite ﬁeld and F ( x ) a simple transcendental extension. Prove that F F ( x ) is a Galois extension. (4) If
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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 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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