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f09_mthsc852_hw13

f09_mthsc852_hw13 - (c Is F isomorphic to a subﬁeld of R...

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MTHSC 851/852 (Abstract Algebra) Dr. Matthew Macauley HW 13 Due Monday, September 21, 2009 (1) Show that [ A : Q ] is infinite. (2) Let F be a field extension of Q such that [ F : Q ] = 2. Prove that there is a unique square-free integer m such that F = Q ( m ). (3) Let S = { m | m N is prime } . (a) Show that Q [ S ] = Q ( S ). (b) Prove that for n N ∪{ 0 } and any choice of n +1 distinct elements a 1 , . . . , a n +1 S , a n +1 Q ( a 1 , . . . , a n ). (c) Deduce that for any finite subset T S we have [ Q ( T ) : Q ] = 2 | T | . Use this fact to argue that [ Q ( S ) : Q ] is infinite. (d) Deduce further that Q ( T ) = Q ( U ), whenever T and U are distinct subsets of S . (4) Let F C be a splitting field for x 3 - 2 over Q . (a) Describe F , i.e. give some generators for a field extension of Q . (b) Compute the dimension of F over Q . Justify your claims.
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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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