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, JUNE 2012 (CUCSSPG2010) 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 Rmodule. 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 questionsEach 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 nonzero 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 Qalgebra 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 questionsEach 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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