2012 University of Calicut Communication B.E. EXTC Fourth Semester M.Sc Degree (Mathematics) examination, University Question paper
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FOURTH SEMESTER M.Sc DEGREE (MATHEMATICS) EXAMINATION,
MT4E02 : ALGEBRAIC NUMBER THEORY
MODEL QUESTION PAPER
Time: 3 hrs. Max. Weightage: 36
(Short Answer Type Questions)
Answer all the questions – Each question has weightage 1
1. Let R be a ring. Define an R-module.
2. Find the minimum polynomial of i + 2 over Q, the field of rationals.
3. Define the ring of integers of a number field K and give the one example.
4. Find an integral basis for Q( 5 )
5. Define a cyclotomic filed. Give one example
6. If K = Q(? ) where 5
? = , find ) (
2 NK ?
7. What are the units in Q( - 3 ).
8. Prove that an associate of an irreducible is irreducible.
9. Define i) The ascending chain condition
ii) The maximal condition
10. If x and y are associates, prove that N(x) = ±N( y)
11. Define : A Euclidean Domain . Give an example.
12. Sketch the lattice in 2 R generated by (0,1) and (1,0)
13. Define the volume v(X) where n X ? R
14. State Kummer's Theorem.
(14 X 1 =14)
(Paragraph Type Questions)
Answer any seven questions-Each question has weightage 2
15. Express the polynomials 2
t +t +t and 3
t in terms of elementary symmetric
polynomials. 16. Prove that the set A of algebraic numbers is a subfield of the complex field C.
17. Find an integral basis and discriminent for Q( d ) if
i) (d -1) is not a multiple of 4
ii) (d -1) is a multiple of 4
18. Find the minimum polynomial of p
= , p is an odd prime , over Q and find its degree.
19. Prove that factorization into irreducibles is not unique in Q( - 26 )
20. Prove that every principal ideal domain is a unique factorization domain.
21. If D is the ring of integers of a number field K, and if a and b are non-zero ideals if D,
then show that N(ab)=N(a) N(b)
22. State and prove Minkowski's theorem.
23. If a a a a n
, , ,............. 1 2 3
is a basis for K over Q, then prove that ) ( ), ( ),......... ( s a1 s a 2 s a n
are linearly independent over R, where s is a Q-algebra homomorphism.
24. Prove that the class group of a number filed is a finite abelian group and the class number
h is finite.
(7 X 2 =14)
(Essay Type Questions)
Answer any two questions-Each question has weightage 4
25. Prove that every subgroup H of a free Abelian group G of rank n is a free of rank s =n .
Also prove that there exists a basis u u u un
, , ,....... 1 2 3
for G and positive integers
a a a a s
, , ,............. 1 2 3
such that a u a u a u a sus
, , ,...... 1 1 2 2 3 3
is a basis for H.
26. a) If K is a number field, Then prove that K = Q(?) for some algebraic number ? .
b) Express Q( ,2 )3 in the form of Q(? )
27. In a domain in which factorization into irreducible is possible prove that each
factorization is unique if and only if every irreducible is prime.
28. Prove that an additive subgroup of n R is a lattice if and only if it is discrete.
(2 X 4 = 8)
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