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Math 422/501, Winter 2006, Term 1
Final Exam
Tuesday, December 12, 2006
No books, notes or calculators
Problem 1. Define the following terms: (a) centre of a group (b) normal subgroup (c) solvable group (d) Sylow subgroup (e) minimal polynomial (f) Galois extension (g) separable polynomial
Problem 2. Carefully state: (a) the orbit equation, (b) a result describing the orbits of an action of the Galois group of the splitting field of a polynomial, (c) a criterion by which to recognize Galois extensions.
Problem 3. Give examples of the following phenomena (without proofs): (a) a non-trivial normal subgroup of the group of isometries of the Euclidean plane, (b) a Sylow 5-subgroup which is not normal, (c) a finite group which is not solvable, (d) a field with 9 elements, (e) a field extension of degree 3 which is not Galois, (f) a field extension of degree 2 which is not Galois, (g) a Galois extension with cyclic Galois group of order 3.
Math 422/501 Dec. 12, 2006
Problem 4. Let (Z/nZ)∗^ be the multiplicative group of the ring Z/nZ. In other words, (Z/nZ)∗ is the group of residue classes a + nZ modulo n, such that a is coprime to n, with multiplication as group operation. Let G be a cyclic group of order n. Prove that Aut(G) is isomorphic to (Z/nZ)∗.
Problem 5. Let G be the group of rotational symmetries of the cube. (a) Prove that G contains 24 elements. (b) How many Sylow 3-subgroups does G contain? Describe them geometrically. (c) How many Sylow 2-subgroups does G contain? Describe them geometrically.
Problem 6. Prove that the fields Q(
- ⊂ R and Q(
- ⊂ R are not isomorphic.
Problem 7. Let K be a field and assume that the characteristic of K does not divide the integer n > 0. Let L/K be a splitting field of the polynomial xn^ − 1 ∈ K[x]. Denote by μn ⊂ L the group of n-th roots of unity in L. (a) Prove that L/K is a Galois extension. Let G be the Galois group. (b) Construct an injective homomorphism of groups G → Aut(μn). (c) Conclude that G is abelian.
Problem 8. Let ζ = e^2 πi/^12 and K = Q(ζ) ⊂ C. (a) Prove that K/Q is a Galois extension. Let G be the Galois group. (b) Prove that G is isomorphic to the Kleinian 4-group. (c) Find the minimal polynomial of ζ over Q. (d) Factor the polynomial x^12 − 1 ∈ Q[x] into irreducible factors.
