Fall 2000

Do all problems

- Let
1 - - >be a short exact sequence of groups such that
*A*- - >*G*- - >*P*- - > 1*A*is abelian, |*P*| = 81, and |*A*| = 332. Show that*G*has a nontrivial center. - Let
*F*be a finite field of odd characteristic. Prove that the rings*F*[*X*]/(*X*^{2}- a) as a ranges over all nonzero elements of*F*fall into exactly two isomorphism classes. - Let
*R*be a finite-dimensional simple algebra and let*M*be a finite-dimensional left*R*-module. Prove that there is a positive integer*d*such that - Let
*B*be a square matrix with rational entries. Show that if there is a monic polynomial*f*[*T*] such that*f*(*B*) = 0 then the trace of*B*is an integer. - Let
*k*be a field. Compute the dimension over*k*of*k*[*X*]/(*X*^{m})*k*[*X*]/(*X*^{n}) - In this problem
*X*,*Y*,*Z*are indeterminates. Define s : (*X*,*Y*,*Z*) - > (*X*,*Y*,*Z*) by s(*h*(*X*,*Y*,*Z*)) =*h*(*Y*,*Z*,*X*) for every rational function*h*in three variables. Prove or disprove: every member of (*X*,*Y*,*Z*) which is left unchanged by s is a rational function of*X*+*Y*+*Z*,*XY*+*YZ*+*XZ*and*XYZ*.