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SECTION - 'A'
Total No. of Questions : 11
[Total No. of Printed Pages : 8
Objective Type Questions
10×1.5=15
IJ-441
M.A./M.Sc. 1st Semester (Reg./Pvt./ATKT)
Examination, 2022-23
Mathematics
Paper - I
Advance Abstract Algebra-I
Time : 3 Hours]
[Maximum Marks : Reg. 85
Pvt. 100
Note :- There are three sections. Attempt all the questions from each section.
SECTION - 'A'
1.
Choose the correct answer :
(i)
An abelian group G has composition series if and only if G is:
(a) Finite
(b) Infinite
(c) Normal
(d) None of these
(a) Finite
(b) Infinite
(c) Normal
(d) None of these
(ii)
A series of any group is called composition series of group if every factor group of series is :
(a) Cyclic
(b) Normal
(c) Simple
(d) None abelian
(a) Cyclic
(b) Normal
(c) Simple
(d) None abelian
(iii)
If a Cyclic gr has exactly one composition series then it is a :
(a) P-group
(b) Solvable
(c) Nilpotent
(d) None of these
(a) P-group
(b) Solvable
(c) Nilpotent
(d) None of these
(iv)
Let N is a normal subgroup of group G then G is solvable group if and only if:
(a) N is solvable
(b) G/N is solvable
(c) Both N and G/N
(d) G/N is normal solvable
(a) N is solvable
(b) G/N is solvable
(c) Both N and G/N
(d) G/N is normal solvable
(v)
Field K is finite extension of field F if:
(a) Degree of F over K is finite
(b) Number of elements of F is finite
(c) [K : F] = finite
(d) [F : K] = finite
(a) Degree of F over K is finite
(b) Number of elements of F is finite
(c) [K : F] = finite
(d) [F : K] = finite
(vi)
If K is finite extension of field F and E is subfield of K containing F then which of the following is correct.
(a) [E : F]/[K : F]
(b) [K : E]/[E : F]
(c) [E : F]/[K : E]
(d) [K : F]/[E : F]
(a) [E : F]/[K : F]
(b) [K : E]/[E : F]
(c) [E : F]/[K : E]
(d) [K : F]/[E : F]
(vii)
If all the extensions of field F are separable then F is called:
(a) Simple Field
(b) Perfect Field
(c) Algebraic Field
(d) Algebraically Closed Field
(a) Simple Field
(b) Perfect Field
(c) Algebraic Field
(d) Algebraically Closed Field
(viii)
If each polynomial in K [x] of positive degree has at least one root in K then K is called:
(a) Algebraic Extension Field
(b) Algebraically Closed Field
(c) Galois Extension Field
(d) Simple Extension Field
(a) Algebraic Extension Field
(b) Algebraically Closed Field
(c) Galois Extension Field
(d) Simple Extension Field
(ix)
If E is finite extension of F then which of the following is correct.
(a) G(E/F) ≤ [E:F]
(b) G(E/F) ≥ [E:F]
(c) G(E/F) = [E:F]
(d) None of these
(a) G(E/F) ≤ [E:F]
(b) G(E/F) ≥ [E:F]
(c) G(E/F) = [E:F]
(d) None of these
(x)
An extension E of a field F is called a Galois extension of F if it is:
(a) Normal Extension
(b) Finite Extension
(c) Separable Extension
(d) All of the above
(a) Normal Extension
(b) Finite Extension
(c) Separable Extension
(d) All of the above
SECTION - 'B'
Sho.t Answer Type Questions 5×5=25
2.
Define subnormal and normal series of group.
OR
Show that a normal subgroup H of G is maximal if and only if G/H is Simple group.
OR
Show that a normal subgroup H of G is maximal if and only if G/H is Simple group.
3.
Define solvable group and Nilpotent Group.
OR
Prove that every abelian group is solvable group.
OR
Prove that every abelian group is solvable group.
4.
Define algebraic element and algebraic extension of field.
OR
Prove that every finite extension of a field F is algebraic extension.
OR
Prove that every finite extension of a field F is algebraic extension.
5.
Define perfect field.
OR
Define algebraically closed field.
OR
Define algebraically closed field.
6.
Define polynomials solvable by radicals.
OR
Define Galois extension of F.
OR
Define Galois extension of F.
SECTION - 'C'
Long Answer Type Questions 5×9=45
7.
Show that any two composition series of a finite group are equivalent.
OR
If G is a Cyclic group such that |G|=PโPโ......Pแตฃ where Pแตข are distinct primes. Show that the number of distinct composition series of G is r!.
OR
If G is a Cyclic group such that |G|=PโPโ......Pแตฃ where Pแตข are distinct primes. Show that the number of distinct composition series of G is r!.
8.
Prove that every finite P-group is Nilpotent group.
OR
Let G be a nilpotent group. Then show that every subgroup of G and every homomorphic image of G are also nilpotent.
OR
Let G be a nilpotent group. Then show that every subgroup of G and every homomorphic image of G are also nilpotent.
9.
Let F ⊆ E ⊆ K be fields such that [K : E] = finite and [E : F] = finites then prove that:
(i) [K : F] = finite
(ii) [K : F] = [K : E] [E : F]
OR
If L is an algebraic extension of K and K is an algebraic extension of F Then show that L is an algebraic extension of F.
(i) [K : F] = finite
(ii) [K : F] = [K : E] [E : F]
OR
If L is an algebraic extension of K and K is an algebraic extension of F Then show that L is an algebraic extension of F.
10.
If E is a finite separable extension of a field F then show that E is a simple extension of F.
OR
Show that K is algebraically closed field if and only if every polynomial in K [x] of positive degree factors completely in K [x] into linear factors.
OR
Show that K is algebraically closed field if and only if every polynomial in K [x] of positive degree factors completely in K [x] into linear factors.
11.
Let E be a finite separable extension of a field. F then show that E is normal extension of F if and only if [E : F] = G(E/F).
OR
What do you understand by insolubility of polynomial equation by radicals prove that general polynomial of degree n ≥ 5 is not solvable by radicals.
OR
What do you understand by insolubility of polynomial equation by radicals prove that general polynomial of degree n ≥ 5 is not solvable by radicals.