#### Abstract Algebra

- Let be a field of characteristic . Let be elements of the field with not a square in . Prove that a necessary and sufficient condition for for some and in is that is a square in . Use this to determine when the field () is biquadratic over .
- Suppose is a rational root of a monic polynomial in . Prove that is an integer.
- Let be a vector space of finite dimension. If is a linear transformation from to prove there is an integer such that the intersection of the image of and the kernel of is .
- Show that and are isomorphic left -modules.
- Show that if and are irreducible -modules, then any nonzero -module homomorphism from to is an isomorphism. Deduce that if is irreducible then is a division ring (this result is called Schur’s Lemma).
- If is a finite abelian group then is naturally a -module. Can this action be extended to make into a -module? Prove or provide a counterexample.
- Prove that is a field which is isomorphic to the complex numbers.
- Prove that the ring contains a subring that is isomorphic to .
- Let and be free groups of finite rank. Prove that if and only if they have the same rank. What facts do you need in order to extend your proof to infinite ranks (where the result is also true)?
- Prove that a finite abelian group is the direct product of its Sylow subgroups.
- For any group define the dual group of (denoted ) to be the set of all homomorphisms from into the multiplicative group of roots of unity of . Define a group operation in by pointwise multiplication of functions. Show that this operation on makes into an abelian group. Moreover, show that if is a finite abelian group, prove that .
- Prove that the group of rigid motions in of an icosahedron is isomorphic to . Prove that the analogous group for the dodecahedron is isomorphic to as well.
- Let be a prime and let be the group of -power roots of in . Prove that the map is a surjective homomorphism. Deduce that is isomorphic to a proper quotient of itself.
- Prove that is not cyclic.
- Let be a group, and consider the bijective map . For which is this map an automorphism?
- Let be a finite fields of prime characteristic with elements. Find the order of , the highest power of dividing the order of , and show that the subgroup of consisting of upper triangular matrices with all diagonal entries is a -Sylow subgroup.
- Let be a prime. Find all abelian groups of order .

#### Linear Algebra

- Prove or give a coutner example: if , , and are subspaces of a vector space such that and , then .
- Prove that if are subspaces of such that , then .
- Let be a finite dimensional vector space and be a subspace of . Show that any linear map on can be extended to a linear map on .
- Prove that if is a linear map on a vector space will null space and range both finite dimensional, then is finite dimensional.
- Let be a linear operator on defined by . Find all eigenvalues and eigenvectors of .
- Suppose is a linear operator and the dimension of the rang of is . Prove that has at most distinct eigenvalues.
- Let and be linear operators on a finite dimensional complex vector space such that . Prove and have a common eigenvector.
- Give an example of an operator whose matrix with respect to some basis contains only nonzero numbers on the diagional, but the operator is not invertible.
- Let be a projection map on a vector space (i.e., ). Prove is the direct sum of the null space and range of .