Do all problems

- Prove that there are no simple groups of order 1755.
- Let
*P*be a finite*p*-group. Prove that every subgroup of*P*appears in some composition series for*P*. - Let
*R*be a principal ideal domain. Let*A*be a finitely generated*R*-module and let*B*be an*R*-submodule of*A*. Assume that there exist nonzero elements*r*and*s*of*R*such that gcd(*r*,*s*) = 1,*rB*= 0, and*s*(*A*/*B*) is a torsion-free*R*-module. Prove that*A*@*B**A*/*B*as*R*-modules. - Let
*F*=**Q**(*i*) and let*K*be the splitting field of*x*^{6}- 7 over*F*.- (a)
- Determine [
*K*:*F*] and write down a basis for*K*over*F*. - (b)
- Show that
Gal(
*K*/*F*) is a dihedral group.

- Let
*R*be a ring with unity 1. Let*P*be a projective*R*-module and let*M*be an*R*-submodule of*P*. Prove: if*P*/*M*is a projective*R*-module, then*M*is a projective*R*-module. - Let
*R*be a commutative Noetherian ring with unity 1. Let*M*be a nonzero*R*-module. Given*m*e*M*, set Ann*m*= {*a*e*R*|*am*= 0} and note that Ann*m*is an ideal of*R*. Prove that there exists*s*e*M*such that Ann*s*is a prime ideal in*R*. (Remember:*R*itself is not a prime ideal in*R*.) - Let
*A*=**C****C**.- (a)
- Prove that there is a well-defined multiplication on
*A*that satisfies the distributive property such that(for all complex numbers*a*_{1}*b*_{1})(*a*_{2}*b*_{2}) =*a*_{1}*a*_{2}*b*_{1}*b*_{2}*a*_{1},*a*_{2},*b*_{1},*b*_{2}. - (b)
- Now assume that this multiplication makes
*A*into a ring. Prove that*A*is not an integral domain.

Peter Linnell 2006-05-22