Algebra

Galois theory and its application

  • What is the degree of Q(\sqrt{2},\sqrt{3},\sqrt{6})?
  • State the field-theoretic condition that determines whether the point (x,y) can be constructed using compass and straight-edge.
  • Which subsets of S_3 can occur as kernels of maps of groups S_3 –> G?
  • Suppose that f: G –> H is a group homomorphism, briefly explain why the ker(f) must be a normal subgroup.
  • Consider the map from S_4 –> S_3 coming from the action of S_4 on partitions of {1,2,3,4} into two 2-element subsets. Exhibit a non-trivial element in the kernel of this map.
  • Give an example of a non-trivial proper subgroup of the dihedral group D {2*4} which is normal and another which is not normal.
  • Consider the action of S_n on ordered pairs of elements of {1, …, n}, what is the stabilizer of (1,2)?
  • Consider the action of S_n on unordered pairs of elements of {1, …, n}, what is the stabilizer of {1,2}?
  • How many elements are in the conjugacy class of (123) in S 4? In A 4?
  • How many elements are in the conjugacy class of (123) in S 5? In A 5?
  • The group Z/3Z acts on the set V of vertices of a cube via rotations around the long diagonal. Describe the orbits of this action. Find subgroups H i such that V is isomorphic to the disjoint union of G/H i as G-sets.
  • Suppose K is a field containing the pth roots of unity, and suppose L:K is a Galois extension with Galois group G, and a is an element of L which is not in K but whose pth power is in K. Describe a nontrivial map G –> Z/pZ.
  • Suppose f: S_n –> Z/pZ is a non-trivial group homomorphism, briefly explain why it follows that p = 2.
  • Suppose f: A n –> Z/pZ is a non-trivial group homomorphism, briefly explain why p = 3.
  • Give an explanation of no more than three sentences explaining the important difference between 4 and 5 that explains why there’s a quartic formula but no quintic formula.
  • Let p be a prime, and z be a primitive pth root of unity, write down all the elemeents of the Galois group \Gamma(Q(z):Q) as automorphisms and describe how they compose.
  • Give an example of a field extension L:K where the Galois correspondence does hold. Your answer should include a subgroup or an intermediate subfield where the correspondence breaks down.