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 π 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)

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 π 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)

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.

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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 π 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 π 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.

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For more details click here or visit http://www.universityofcalicut.info