Do all problems

- Let
*G*be the group of upper triangular invertible 3×3 matrices over the field 𝔽_{5}of 5 elements.- (a)
- Show that
*G*has a unique Sylow 5-subgroup*P*_{5}. - (b)
- Construct explicitly a composition series for
*P*_{5}. - (c)
- Show that
*G*is isomorphic to the semidirect product*P*_{2}⋉*P*_{5}.

- Let
*S*be the ring of*n*×*n*-matrices with entries in a field*F*.- (a)
- Show that the
*S*-module*V*= 𝔽^{n}of column vectors is a simple left*S*-module (simple means it is nonzero and has no submodules other than 0 and itself). - (b)
- Show that every left ideal of
*S*is a projective left*S*-module.

- Find the Galois group of the polynomial
*f*(*x*) = (*x*^{12}-1)(*x*^{2}- 2*x*+ 2) over ℚ. - Classify the conjugacy classes of
5×5 matrices of order 3
- (a)
- with coefficients in
ℚ.
- (b)
- with coefficients in ℂ.

- Let
*M*be a module over the integral domain*R*.- (a)
- Prove directly that
*M*= 0 if and only if*M*_{P}= 0 for all prime ideals*P*, where*M*_{P}is the localization of*M*at*P*. - (b)
- Prove that an
*R*-module homomorphism*f*:*M*→*N*is surjective if and only if, for all prime ideals*P*, the maps*f*_{P}:*M*_{P}→*N*_{P}are surjective, where by definition*f*_{P}(*m*/*d*)=*f*(*m*)/*d*for all*m*∈*M*and*d*∈*R*\*P*.

- Let
*G*be a group of order 2^{a}*p*, where 1≤*a*≤3 and*p*≥3 is a prime. Prove that*G*cannot be simple. - Let
*F*= (*F*_{1},...,*F*_{m}) be a system of*m*polynomial equations, where each*F*_{i}∈ℤ[*x*_{1},...,*x*_{r}]. Consider the following statements:- (a)
- The system
*F*has solutions in ℤ. - (b)
- The system
*F*has solutions in ℤ/*n*ℤ for any*n*≥1. - (c)
- The system
*F*has solutions in ℤ/*p*^{s}ℤ where*p*is any prime and*s*≥1. - (d)
- The system
*F*has solutions in ℤ/*p*ℤ for any prime*p*.

Prove that each statement implies the next. Prove that (c) implies (b) and give counterexamples to the other five backward implications (d) implies (c), (d) implies (b), etc.

Peter Linnell 2016-01-09