M.C.A. DEGREE EXAMINATION - MODULES AND GALOISTHEORY

[ganeshbala]

ADVANCEDCOURSE - II

MODULES AND GALOISTHEORY

Time: Three hours Maximum: 100 marks

Answer any FIVE questions.All questions carry equal marks.


1. (a) State and prove fundamental theorem for
homomorphism of rings.
(b) Show that an ideal M of a ring R is
maximal if and only if ~ is a field.


2. (a) Show that any two elements a and b in a
Euclidean ring R have a greatest common divisor.
(b) Let F be a field. Prove that the ring of
polynomials F

• over F is an Euclidean ring.
  
  
  3. (a) State and prove Eisenstein irreducibility
  criterion.
  (b) If R is a unique factorization domain, then
  show that R
• is also a unique factorization domain.
  
  
  4. (a) Define a generating set in a vector space.
  Show that {VI'V2'''. vn} is a minimal generating set of
  a vector space V if and only ifit is a basis of V .
  (b) If VI and V2 are subspaces of a vector space
  V , then prove that
  dim {VI+V2) =dim VI + dim V2 - dim (VI n V2).
  
  
  5. (a) If V and Ware vector spaces of dimensions
  m and n respectively over F, then show that
  Horn (V, W) is of dimension mn over F .
  (b) If V is a finite dimensional vector space and
  v =I:0 is in V , prove that there is an element rEV (V
  is the dual of V), such that r(v) =I:O.
  
  
  6. (a) Show that m an
  M =MI E9M2 E9... E9Mn if and only if
  R -module,
  (i) M =MI +M2 +... +Mn
  (ii) Mi n(MI +M2 +... Mi-I +Mi+1 +
  .., +M n) =(0) for alIi, 1 S;is; n.
  (b) Let M be a finitely generated module over a
  principal ideal domain R. Show that M can be
  expressed as M = F EBt(M), where F is free.
  
  
  7. (a) Let Fe K c L be field extensions slwh tha
  K/F and L/K are finite. Then prove that L/F is finit,
  and [L: F] =[L: K] [K: F].
  (b) Let K be an extension of a field Fane
  a EK . Then show that a is algebraic over F if anc
  only if F(a) is a finite extension of F .
  
  
  
  8. (a) Let f(x) be any polynomial of degree n ~1
  over a field F . Prove that there is an extension K of F
  I
  '
  .
  .
  of degree at most n! in which f(x) has n roots. .
  (b) Obtain a splitting field of X4 - 2 over Q. ,I
  
  
  
  9. State and prove the fundamental theorem of
  Galois theory. ~
  
  
  10. (a) Prove that the field of complex numbers'
  algebraically dosed.
  (b) Show that for every prime number p ani!
  n ~ 1, there exists a field with p' element.. - I