Do all problems

- Prove that there is no simple group of order 380.
- Let
*k*be a field with |*k*| = 7, let*n*be a positive integer, and let*f*,*g*be coprime polynomials in*k*[*x*_{1},...,*x*_{n}]. If*f*^{3}-*g*^{3}=*h*^{3}for some nonzero polynomial*h*∈*k*[*x*_{1},...,*x*_{n}], prove that there exists*p*∈*k*[*x*_{1},...,*x*_{n}] and*u*∈*k*such that*f*-*g*=*up*^{3}. Hint: factor*f*^{3}-*g*^{3}as a product of three polynomials, and note that these polynomials are pairwise coprime. - Let
*R*be a PID, let*p*be a prime in*R*, and let*M*,*N*be finitely generated left*R*-modules such that*pM*≌*pN*. Assume that if 0≠*m*∈*M*or*N*and*pm*= 0, then*Rm*is not a direct summand of*M*or*N*respectively (i.e. there is no submodule*X*such that*Rm*⊕*X*=*M*or*N*). Prove that*M*≌*N*. - Let
*f*(*x*)∈**Q**[*x*] be an irreducible polynomial of degree 9, let*K*be a splitting field for*f*over**Q**, and let α∈*K*be a root of*f*. Suppose that [*K*:**Q**] = 27. Prove that**Q**(α) contains a field of degree 3 over**Q**. - Let
*p*be a prime and let*A*denote all*p*^{n}-th roots of unity in**C**. Thus*A*is the abelian subgroup of the nonzero complex numbers under multiplication defined by {*e*^{2πim/pn}|*m*,*n*∈**N**}, in particular*A*is a**Z**-module. Determine*A*⊗_{Z}*A*. - Let
*k*be a field, let*A*∈M_{3}(*k*), the 3 by 3 matrices with entries in*k*, and suppose the characteristic polynomial of*A*is*x*^{3}. Prove that*A*has a square root, that is a matrix*B*∈M_{3}(*k*) such that*B*^{2}=*A*, if and only if the minimal polynomial of*A*is*x*or*x*^{2}. - Let
*R*be an integral domain, let*n*be a positive integer, let*S*be a subset of the polynomial ring in*n*variables*R*[*x*_{1},...,*x*_{n}], and define*Z*(*S*) = {(*r*_{1},...,*r*_{n})∈*R*^{n}|*f*(*r*_{1},...,*r*_{n}) = 0 for all*f*∈*S*}, the zero set of*S*. Prove that there exists a finite subset*T*of*S*such that*Z*(*S*) =*Z*(*T*).

Peter Linnell 2011-08-13