**Algebra Preliminary Exam, Spring 1980**

- Let
*A*,*B*and*C*be finite abelian groups. If , prove that . - Show that there exists no simple group of order 56.
- Let
*T*denote the set of all matrices with eigenvalues 4,4,17,17,17. Define a relation on*T*by if*M*_{1}and*M*_{2}are similar matrices. How many equivalence classes does*T*have? Justify your answer. (Assume that the matrices are over .) - Give an example of a unique factorization domain (UFD) which is not a principal ideal domain (PID).
- What is the Galois group of
*x*^{3}- 10 over ? Find all normal subfields of the splitting field. - Recall: if
*E*is the splitting field of a polynomial*f*over*F*, then is called the Galois group of*f*over*F*. The Galois group of*f*over*F*is said to be transitive if given any two roots*r*_{1}and*r*_{2}of*f*in*E*, there exists in with .- Prove that if
*f*is a separable irreducible polynomial, then the Galois group of*f*is transitive. - Show that even though the Galois group of
*f*is transitive, not every permutation of the roots need occur. (Hint: consider*x*^{4}-2 over .)

- Prove that if
- Let
*A*be a local ring with maximal ideal , let*k*be , and let*M*be a finitely generated*A*-module. Show that if , then*M*= 0. (Hint: use Nakayama's lemma.)

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Thu Aug 1 09:39:08 EDT 1996