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 M1 and M2 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 x3 - 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 r1 and r2 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
x4-2 over .)
- 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
Thu Aug 1 09:39:08 EDT 1996