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St. Joseph’s College of Arts & Science (Autonomous) St. Joseph’s College Road, Cuddalore – 607001 PMT913 - APPLIED ABSTRACT ALGEBRA

Time : 3 hrs

Max Marks :75 SECTION – A (5X2=10) Answer ALL Questions

1. Define a chain. 2. Define a tautology. 3. Define the Mobius function. 4. If f ∈ Fq [ x ] be an irreducible polynomial over Fq of degree k . Then prove that ord f divides q k − 1 . 5. Define orthogonal. SECTION – B (3X5=15) Answer any THREE Questions 6. Let ( L, ≤) be a lattice ordered set. If we define

x ∧ y : = inf( x, y ), x ∨ y := sup( x, y ), then show that ( l , ∧, ∨ ) is an algebraic lattice.

7. Let A, B, C, D be sets. Simplify M = ( A I ( B \ C ) ) U ( B \ C ) U ( ( D \ A) I ( D \ B ) I C ) ,

where B∆C denotes the symmetric difference ( B \ C ) U ( C \ B ) of B and C.

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8. Let F be a finite field with q elements. Then prove that the followings: (i) The multiplicative group ( F *, ⋅) of the nonzero elements of F is cyclic of order q − 1 (ii) All elements a of F satisfy a q − a = 0 . 9. Prove that an irreducible polynomial f over Fq of degree k divides x q − x if and only if k divides n . n

10. Prove that a linear code C Í Vn is cyclic if and only if C is an ideal in Vn . SECTION- C (5X10=50) Answer ALL Questions 11. a) State and prove Representation Theorem. (10) (or) b) Let B be a Boolean algebra and let I be a nonempty subset of B . Then show that the following conditions are equivalent: (i) I < B; (ii) For all i, j ∈ I and b ∈ B : i + j ∈ I and b ≤ i ⇒ b ∈ I ; (iii) I is the kernel of a Boolean homomorphism from B into another Boolean algebra. (10) 12. a) Determine if the following argument is correct: “If the workers in a company do not go on strike, then a necessary and sufficient condition for salary increases is that the hours of work increase. In case of a salary increase there will be no strikes. If working hours increase, there will be no salary increase. Therefore salaries will not be increased.” The compound statement can be formulated as follows:

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(( ¬S → (( I → W ) ∧ (W → I ))) ∧ ( I → ¬S ) ∧ (W → ¬I ) → ¬I ) , where S denotes “strike”, I “salary increase”, and W “increase of working hours”. The abbreviate compound statement by ( (1) ∧ ( 2 ) ∧ ( 3) ) → ¬I . (10) (or) b) Let (A, P )be a probability space. Then show that the followings for all b1 , b2 , b Î A . (i) b1 , b2 = 0 Þ P (b1 + b2 )= P (b1 )+ P (b2 ); (ii) P (b1 + b2 )= P (b1 )+ P (b2 )- P (b1b2 ); (iii) b1 £ b2 Þ P (b1 )£ P (b2 ); (iv)

P (b)Î [0,1];

(v) P (b ')= 1- P (b).

(10)

13. a) Let p be a prime and let m, n be natural numbers. Then prove that the followings: (i) If Fp m is a subfield of F p n , then m n . (ii) If m n , then Fp m

F pn .There is exactly one subfield of

F pn with p m elements. (or) b) State and prove Mobius Inversion Formula.

(10) (10)

14. a) i) Prove that if f Î Fq [x]be a polynomial of degree m ³ 1 with f (0) ¹ 0 . Then there exists a positive integer e £ q m - 1 such that f divides x e .

ii) Let f Î Fq [x]be an irreducible polynomial over Fq of degree m ³ 2 . Then prove that ord f is equal to the order of any root f in Fq*m . (5+5)

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(or) b) Prove that a polynomial f Î Fq [x]of degree m is primitive if and only if f is monic, f (0) ¹ 0 , and the order of f is equal to qm - 1. (10) 15. a) State and prove Gilbert-Varshamov Bounded theorem. (10) (or) b) Prove that let C be an ideal ¹ {0}of Vn . Then there exists a unique g Î Vn with the following properties: (i) g x n - 1 in Fq [x ]; (ii) C = (g ); (iii) g is monic.

Hence every ideal of Vn is principal (note that Vn is not an integral domain, however).

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(10)