Unformatted text preview: Q [ x ] is a UFD. Let r = f / g ∈ Q ( x ) (the ﬁeld of fractions of Q [ x ] ) with f , g ∈ Q [ x ] \ 0, coprime (no common irreducible factor), and at least one of f , g has degree nonzero. (a) Prove that there is a unique monomorphism θ r : Q ( x ) → Q ( x ) such that θ r ( x ) = r . (b) If f and g both have degree at most 1, prove that θ r is an automorphism. Hint: if r = ax + b cx + d , consider the inverse of the matrix ± a b c d ² and use this to construct s ∈ Q ( x ) such that θ r θ s ( x ) = θ s θ r ( x ) = x . (4 points) 5. Section 8.3, Exercise 2 on page 293. (2 points) (5 problems, 13 points altogether)...
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 Fall '07
 PALinnell
 Math, Algebra, Ring, Prime number, Ring theory, Commutative ring

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