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Qualifying Exam in Algebra - January 1992, Exams of Algebra

Punjab Engineering CollegeAlgebra

A qualifying exam in algebra from january 1992. The exam covers group theory, ring theory, and field theory. It includes 18 problems that test the student's understanding of various concepts in algebra, such as centralizers, homomorphisms, maximal subgroups, normal subgroups, inner automorphisms, simple groups, matrix operations, prime ideals, unique factorization domains, extension fields, minimal polynomials, galois groups, and dual spaces.

Typology: Exams

2012/2013

Uploaded on 02/23/2013

mohan_das
mohan_das 🇮🇳

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QUALIFYING EXAM IN ALGEBRA
JANUARY 1992
1. Work as many problems as you can. It is to your advantage to demonstrate a broad
background.
2. If you feel there is a misprint or error in the statement of the problem, then interpret
it in such a way that the problem is not trivial.
1
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QUALIFYING EXAM IN ALGEBRA

JANUARY 1992

  1. Work as many problems as you can. It is to your advantage to demonstrate a broad background.
  2. If you feel there is a misprint or error in the statement of the problem, then interpret it in such a way that the problem is not trivial.

Group Theory

  1. (a) Find the centralizer in S 7 of ( 1 2 3 )( 4 5 6 7 ). (b) How many elements of order 12 are there in S 7?
  2. Let f : G → H be a homomorphism of groups with kernel K and image I. (a) Show that if N is a subgroup of G then f −^1 (f (N )) = KN. (b) Show that if L is a subgroup of H then f (f −^1 (L)) = I ∩ L.
  3. Let G be a finite group. (a) Show that every proper subgroup of G is contained in a maximal subgroup. (b) Show that the intersection of all maximal subgroups of G is a normal subgroup.
  4. Let N be a group with trivial center such that all automorphisms of N are inner automorphisms. Show that whenever N occurs as a normal subgroup of a group G, there is a subgroup H of G such that G = H × N.
  5. Let G be a subgroup of the symmetric group Sn. Show that if G contains an odd permutation then G ∩ An is of index 2 in G.
  6. Show that a simple group of order 168 must be isomorphic to a subgroup of the alternating group A 8.

Field Theory

  1. Let K be an extension field of the field F such that [K : F ] is odd. Show that if u ∈ K then F (u) = F (u^2 ).
  2. Let F ⊂ E ⊂ K be a tower of fields such that K = F (α) with α algebraic over F. Prove that if f (x) ∈ F [x] is the minimal polynomial of α over F and F 6 = E, then f (x) is not irreducible in E[x].
  3. Let f (x) ∈ Q[x] be an irreducible polynomial of degree n with roots α 1 ,... , αn. Show that ∑^ n i= α^1 i is a rational number.
  4. Let f (x) = x^4 + x^3 + 4x − 1 ∈ Z 5 [x]. Find the Galois group of the splitting field of f over Z 5.
  5. Let η be a complex primitive 11-th root of unity and let K = Q(η). Find Gal(K/Q) and express each intermediate field F between K and Q as F = Q(β) for some β ∈ K.
  6. Let f (x) ∈ Q[x] be an irreducible polynomial of degree 4 with exactly 2 real roots. Show that the Galois group of f is either S 4 or the dihedral group of order 8.

Linear Algebra

  1. Let V and W be finite dimensional vector spaces and let T : V → W be a linear transformation. Show that dim(ker T ) + dim(Im T ) = dim(V ).
  2. Let V be a finite dimensional vector space over the field F. Let V ∗^ be the dual space of V (that is, V ∗^ is the vector space of linear transformations T : V → F ). Show that V ∼= V ∗.