Skip to main content

M.Sc MATHEMATICS MODEL QUESTION PAPER

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)

 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. 

Click here to download Question paper


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

Comments

Popular posts from this blog

B.SC 2ND YEAR CHEMISTRY QUESTION PAPERS

KUVEMPU UNIVERSITY SECOND YEAR B.SC CHEMISTRY EXAMINATION QUESTION PAPERS (FROM 2008 TO 2020)  Published by  www.questionpapersdownload.com   BSc II Year - Chemistry Paper-II_2016 - Download BSc II Year - Chemistry Paper-II_2017 - Download BSc II Year - Chemistry Paper-II_2014 - Download All other 2nd Year B.Sc Nursing Question Papers avaialble   Click here to download/view more B.Sc Nursing Question Papers 2010-2020 Chemistry Paper-II 2008 2009 2010 2011 2012 2013 2014  

PDO PREVIOUS YEAR QUESTION PAPERS

The PDO (Panchayath Development Officer) and Panchayath Secretary Examinations will be conducted by Karnataka Examination Authority. The Online registrations will be starting from 16/09/2016. Most candidates are requested us to provide previous question papers of PDO Examination. The Question papers of PDO Examination conducted in the year 2011 is published here. For more PDO Exam Questions Click here PART 1 COMPUTER KNOWLEDGE QUESTIONS PDO Exam Question Paper Page 2 PDO Exam Question Paper Page 3 PDO Exam Question Paper Page 4 PDO Exam Question Paper Page 5 CLICK HERE FOR PDO QUESTION PAPER PAGE NO. 01-05  PART II RURAL DEVELOPMENT AND PANCHAYAT RAJ QUESTIONS - PAGE NO. 6-10 CLICK HERE FOR PDO QUESTION PAPER PAGE NO. 11-14- General Kannada CLICK HERE FOR PDO QUESTION PAPER PAGE NO. 15-19- General English CLICK HERE FOR PDO QUESTION PAPER PAGE NO. 20-23- General Knowledge MORE PDO EXAM QUESTIONS Click here to Join Our Facebook group to ge

PRIMARY HEALTH CARE QUESTION PAPER

ANM (AUXILIARY NURSING MIDWIFERY)MODEL QUESTION PAPER PAPER-III Primary Health Care Question Paper Published by  http://www.questionpapersdownload.com ANSWER ALL QUESIONS 3 long Questions of 15 Marks each (3X15) -45 Short Note Variety (15 Marks) Objective Type Questions (Fill in Blanks / True & False/Abbreviations) -15Marks Q.1. a) What is the concept of occurance of disease ? (5+5+5) b) What is illness? c) Desccribe the classification of disease? or a) What do you mean by Incubation Period ? b) Write the clarification of Micro Orgnism ? c) Write the factors affecting the growth & destruction of organism? Q2. a) Define bed sore? (5+4+6) b) What are the causes of bed sore ?