Algebra Exercises

Abstract Algebra

  • Let F be a field of characteristic \neq 2. Let a,b be elements of the field F with b not a square in F. Prove that a necessary and sufficient condition for \sqrt{a+\sqrt{b}} = \sqrt{m}+\sqrt{n} for some m and n in F is that a^{2}-b is a square in F. Use this to determine when the field \mathbb{Q}(\sqrt{a+\sqrt{b}}) (a,b \in \mathbb{Q}) is biquadratic over \mathbb{Q}.
  • Suppose \alpha is a rational root of a monic polynomial in \mathbb{Z}[x]. Prove that \alpha is an integer.
  • Let V be a vector space of finite dimension. If \varphi is a linear transformation from V to V prove there is an integer m such that the intersection of the image of \varphi^{m} and the kernel of \varphi^{m} is {0}.
  • Show that \mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{Q} and \mathbb{Q} \otimes_{\mathbb{Q}} \mathbb{Q} are isomorphic left \mathbb{Q}-modules.
  • Show that if M_{1} and M_{2} are irreducible R-modules, then any nonzero R-module homomorphism from M_{1} to M_{2} is an isomorphism. Deduce that if M is irreducible then \text{End}_{R}(M) is a division ring (this result is called Schur’s Lemma).
  • If M is a finite abelian group then M is naturally a \mathbb{Z}-module. Can this action be extended to make M into a \mathbb{Q}-module? Prove or provide a counterexample.
  • Prove that \mathbb{R}[x]/(x^{2}+1) is a field which is isomorphic to the complex numbers.
  • Prove that the ring M_{2}(\mathbb{R}) contains a subring that is isomorphic to \mathbb{C}.
  • Let F_{1} and F_{2} be free groups of finite rank. Prove that F_{1} \cong F_{2} 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 G define the dual group of G (denoted \hat{G}) to be the set of all homomorphisms from G into the multiplicative group of roots of unity of \mathbb{C}. Define a group operation in \hat{G} by pointwise multiplication of functions. Show that this operation on \hat{G} makes \hat{G} into an abelian group. Moreover, show that if G is a finite abelian group, prove that \hat{G} \cong G.
  • Prove that the group of rigid motions in \mathbb{R}^{3} of an icosahedron is isomorphic to A_{5}. Prove that the analogous group for the dodecahedron is isomorphic to A_{5} as well.
  • Let p be a prime and let G be the group of p-power roots of 1 in \mathbb{C}. Prove that the map z \mapsto z^{p} is a surjective homomorphism. Deduce that G is isomorphic to a proper quotient of itself.
  • Prove that \mathbb{Q} \times \mathbb{Q} is not cyclic.
  • Let G be a group, and consider the bijective map g \mapsto g^{-1}. For which G is this map an automorphism?
  • Let \mathbb{F}_{q} be a finite fields of prime characteristic p with q elements. Find the order of GL_{n}(\mathbb{F}_{q}), the highest power of p dividing the order of GL_{n}(\mathbb{F}_{q}), and show that the subgroup of GL_{n}(\mathbb{F}_{q}) consisting of upper triangular matrices with all diagonal entries 1 is a p-Sylow subgroup.
  • Let p be a prime. Find all abelian groups of order p.

Linear Algebra

  • Prove or give a coutner example: if U_{1}, U_{2}, and W are subspaces of a vector space V such that V = U_{1} \oplus W and V = U_{2} \oplus W, then U_{1} = U_{2}.
  • Prove that if U_{1},\ldots,U_{m} are subspaces of V such that V = U_{1} \oplus \cdots \oplus U_{m}, then dim(V) = dim(U_{1})+\cdots+dim(U_{m}).
  • Let V be a finite dimensional vector space and U be a subspace of V. Show that any linear map T on U can be extended to a linear map on V.
  • Prove that if T is a linear map on a vector space V will null space and range both finite dimensional, then V is finite dimensional.
  • Let T be a linear operator on \mathbb{C}^{2} defined by T(w,z) = (z,w). Find all eigenvalues and eigenvectors of T.
  • Suppose T is a linear operator and the dimension of the rang of T is k. Prove that T has at most k+1 distinct eigenvalues.
  • Let S and T be linear operators on a finite dimensional complex vector space such that ST = TS. Prove S and T 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 P be a projection map on a vector space V (i.e., P^{2} = P). Prove V is the direct sum of the null space and range of P.