Algebra Preliminary Exam, Spring 1980

1. Let A,B and C be finite abelian groups. If , prove that .
2. Show that there exists no simple group of order 56.
3. Let T denote the set of all matrices with eigenvalues 4,4,17,17,17. Define a relation on T by if M₁ and M₂ are similar matrices. How many equivalence classes does T have? Justify your answer. (Assume that the matrices are over .)
4. Give an example of a unique factorization domain (UFD) which is not a principal ideal domain (PID).
5. What is the Galois group of x³ - 10 over ? Find all normal subfields of the splitting field.
6. 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 r₁ and r₂ of f in E, there exists in with .
   1. Prove that if f is a separable irreducible polynomial, then the Galois group of f is transitive.
   2. Show that even though the Galois group of f is transitive, not every permutation of the roots need occur. (Hint: consider x⁴-2 over .)
7. 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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