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(ζ ) where 5
2 i
e
Ï€
ζ = , find ) (
2 NK ζ
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 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€
ξ
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 α α α α n
, , ,…………. 1 2 3
is a basis for K over Q, then prove that ) ( ), ( ),……… ( σ α1 σ α 2 σ α n
are linearly independent over R, where σ 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
α α α α s
, , ,…………. 1 2 3
such that α u α u α u α 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(θ) for some algebraic number θ .
b) Express Q( ,2 )3 in the form of Q(θ )
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(ζ ) where 5
2 i
e
Ï€
ζ = , find ) (
2 NK ζ
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 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€
ξ
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 α α α α n
, , ,…………. 1 2 3
is a basis for K over Q, then prove that ) ( ), ( ),……… ( σ α1 σ α 2 σ α n
are linearly independent over R, where σ 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
α α α α s
, , ,…………. 1 2 3
such that α u α u α u α 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(θ) for some algebraic number θ .
b) Express Q( ,2 )3 in the form of Q(θ )
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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