Q8/13E/04-12 Reg. No

St. Joseph’s College of Arts & Science (Autonomous) St. Joseph’s College Road, Cuddalore – 607001 PMT806S - ALGEBRA - II Time : 3 hrs Max Marks :75

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SECTION – A (5X2=10) Answer ALL Questions 1. Define algebraic number. 2. Define splitting field. 3. Prove that the fixed field of G is a subfield of K. 4. Define commentator group. 5. Define algebraic over a field SECTION – B (3X5= 15) Answer any THREE Questions 6. If a,b is K are algebraic over F, then prove that

a ± b, ab and

a ( b ≠ 0 ) are algebraic over F. b

7. Prove that if f ( x) ∈ F [ x] of degree n ≥ 1 then there is an extension E of F of degree at most n! is which f ( x) has n roots.

8. If K is a finite extension of F, then prove that G (K,F) is a finite group then prove that O ( G ( K , F ) ) ≤ [ K : F ] . 9. If F has all its n th root of unity and a ≠ o in F, xn − a ∈ F [ x] and K P

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be the splitting field of F, then prove that.

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Q8/13E/04-12 i) K = F ( u ) where u is any root of xn − a . ii) The Galois gorsy of xn − a over F is a belian. 10. For all x, y ∈ Q , prove that N ( xy ) = N ( x) N ( y ) . SECTION – C (5X10=50) Answer ALL Questions 11. a) If L is a finite extension of K and if K is a finite extension of F, then prove that L is a finite extension of F. (or) b) Prove that the element a ∈ k is algebraic over F iff F(a) is a finite extension of F. 12. a) Prove that any splitting fields E and E ’ of the polynomials f ( x) ∈ F [ x] and f ' ( t ) ∈ F ' [ t ] , respectively, are isomorphic by P

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an isomorphism φ with the property that αφ = α 1 for every α ∈F . (or) b) Prove that a polynomial of degree n over a field can have at most n roots in any extension field. 13. a) If F is of characteristic O and if a,b are algebraic over F, then prove that there exists an element C ∈ F ( a , b ) such that

F ( a, b) = F ( c) .

(or) b) State and prove that fundamental theorem of Galois theory. 14. a) If p ( x) ∈ F [ x] is solvable by radicals over F, then prove that the Galois group over F of p ( x) is a solvable group.

(or) b) State and prove wedderburn theorem.

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Q8/13E/04-12 15. a) State and prove the Left – division algorithon Lemma. (or) b) i) Prove that the adjoint in Q satisfies i) x∗∗ = x ∗ ii) ( δ x + ν y ) = δ x∗ + ν y∗ iii) ( xy ) = y∗ x∗ ∗

ii) If C is the field of complex numbers and the division ring D is algebraic over C, then prove that D=C.

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