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( − 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 polyn