Algebra2010jan - A is a normal lower triangular matrix over...

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NAME: Algebra Ph.D. Preliminary Exam January 12, 2010 1. Recall that a subgroup H of a group G is called characteristic if φ ( H ) H for every automorphism φ of G . (a) Prove that characteristic subgroups are always normal. (b) Let P be a p -Sylow subgroup of a finite group G and assume that P is normal in G . Prove that P is a characteristic subgroup of G . 2. Prove that there are no simple groups of order 20 or 57. 3. Let G be an abelian group of order n and assume that G has at most one subgroup of order d for each d | n . Prove that G is a cyclic group. 4. Let R be a commutative ring such that the polynomial ring R [ X ] is a PID. Prove that R is a field. 5. Recall that a n × n matrix A is normal if AA * = A * A . Prove that if
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Unformatted text preview: A is a normal lower triangular matrix over the complex numbers, the A is a diagonal matrix. 6. Let R = Z [ X ] and let I = (2 ,X ). Prove that I is not a free R-module but it is torsion free. 7. Let F be a nite eld. Prove that the product of the non zero elements of F is-1. 8. Let = p 2 + 2. Find the minimal polynomial of over Q and show that = p 2- 2 is another root of this minimal polynomial. Show that the degree of Q ( ) over Q is 4. Prove that sending to = p 2- 2 is an automorphism of Q ( ) over Q . Describe the Galois group of Q ( ) over Q . 1...
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