Algebra Qualifying Exam, Fall 1989

Do six problems

1. Compute the Galois group of 3x² + 7x + 21 over

   

2. By an ``N-group", we mean a finite group with the property that every nonidentity homomorphic image has a nonidentity center. Prove that maximal subgroups of N-groups are always normal.
3. Let R be an integral domain and let K be its field of fractions. Assume that if , then either or . Prove that
   1. R is a local ring.
   2. R is integrally closed in K.
4. Let R be a PID and let A, M be nonzero finitely generated R-modules.
   1. Show that if A is torsion free, then .  
   2. Provide a counterexample to the conclusion of (1) in the case A is not torsion free.
5. Assume that p and q are distinct primes. Show that a group of order p²q cannot be simple.
6. Let k be a field. If , define

   

   Prove that if f₁,f₂, ... is a countable list of polynomials in k[X₁, ... ,X_n], then there is a positive integer T such that

   

7. Let k be a field. Prove that if A and B are two -matrices with entries in k, both of which have minimal polynomial X^n-1, then A and B are similar.

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