12-18-2014, 08:53 PM

Here are the previous year question paper from calicut university. This is the original question paper from M.Sc Mathematics fourth sememster examination conducted by calicut university in the year 2012. Certainly download question paper from here and use it to prepare for your upcoming exam.

FOURTH SEMESTER M.Sc DEGREE (MATHEMATICS) EXAMINATION,

JUNE 2012

(CUCSS-PG-2010)

MT4E02 : ALGEBRAIC NUMBER THEORY

MODEL QUESTION PAPER

Time: 3 hrs. Max. Weightage: 36

PART A

(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

2 i

e

p

? = , 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)

PART B

(Paragraph Type Questions)

Answer any seven questions-Each question has weightage 2

15. Express the polynomials 2

3

2

2

2

1

t +t +t and 3

1

t +

3

2

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

i

e

p

?

2

= , 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)

PART –C

(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)