Unformatted text preview: (c) Is F isomorphic to a subﬁeld of R ? Prove or disprove. (5) Let a 1 ,a 2 ∈ C be any two numbers which are algebraic over Q . Show that if there exists an isomorphism φ : Q ( a 1 ) → Q ( a 2 ), leaving Q elementwise ﬁxed and satisfying φ ( a 1 ) = a 2 , then a 1 and a 2 have the same minimal polynomial over Q . (6) Fix a ﬁeld F , and deﬁne a category C F whose objects are the extension ﬁelds of F . Deﬁne morphisms in C F in such a fashion that the algebraic closure ¯ F of F arises as the solution of a universal mapping problem. Carefully formulate this problem and prove all of your claims....
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 Fall '08
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 Algebra, Vector Space, Category theory, Dr. Matthew Macauley, Matthew Macauley HW

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