FOURTH SEMESTER M.Sc DEGREE (MATHEMATICS) EXAMINATION,

JUNE 2012

(CUCSS-PG-2010)

MT4E02 : ALGEBRAIC NUMBER THEORY

MODEL QUESTION PAPER

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(Î¶ ) where 5

2 i

e

Ï€

Î¶ = , find ) (

2 NK Î¶

7. What are the units in Q( âˆ’ 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) = Â±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 âŠ‚ R

14. State Kummerâ€™s Theorem.

(14 X 1 =14)

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 forQ( 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

Ï€

Î¾

2

= , p is an odd prime , over Q and find its degree.

19. Prove that factorization into irreducibles is not unique in Q( âˆ’ 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 Minkowskiâ€™s theorem.

23. If Î± Î± Î± Î± n

, , ,…………. 1 2 3

is a basis for K over Q, then prove that ) ( ), ( ),……… ( Ïƒ Î±1 Ïƒ Î± 2 Ïƒ Î± n

are linearly independent over R, where Ïƒ 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)

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 â‰¤n .

Also prove that there exists a basis u u u un

, , ,……. 1 2 3

for G and positive integers

Î± Î± Î± Î± s

, , ,…………. 1 2 3

such that Î± u Î± u Î± u Î± 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(Î¸) for some algebraic number Î¸ .

b) Express Q( ,2 )3 in the form of Q(Î¸ )

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.

Click here to download Question paper

JUNE 2012

(CUCSS-PG-2010)

MT4E02 : ALGEBRAIC NUMBER THEORY

MODEL QUESTION PAPER

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(Î¶ ) where 5

2 i

e

Ï€

Î¶ = , find ) (

2 NK Î¶

7. What are the units in Q( âˆ’ 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) = Â±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 âŠ‚ R

14. State Kummerâ€™s Theorem.

(14 X 1 =14)

**PART B**

(Paragraph Type Questions)(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 forQ( 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

Ï€

Î¾

2

= , p is an odd prime , over Q and find its degree.

19. Prove that factorization into irreducibles is not unique in Q( âˆ’ 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 Minkowskiâ€™s theorem.

23. If Î± Î± Î± Î± n

, , ,…………. 1 2 3

is a basis for K over Q, then prove that ) ( ), ( ),……… ( Ïƒ Î±1 Ïƒ Î± 2 Ïƒ Î± n

are linearly independent over R, where Ïƒ 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 â€“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 â‰¤n .

Also prove that there exists a basis u u u un

, , ,……. 1 2 3

for G and positive integers

Î± Î± Î± Î± s

, , ,…………. 1 2 3

such that Î± u Î± u Î± u Î± 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(Î¸) for some algebraic number Î¸ .

b) Express Q( ,2 )3 in the form of Q(Î¸ )

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.

Click here to download Question paper

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