Advance Abstract Algebra-I

A series {e} = G₀ ⊆ G₁ ⊆ G₂ ⊆ ......... ⊆ Gₙ = {G} of group G is said to be composition series if :
(a) Each Gᵢ₊₁/Gᵢ is Cyclic group
(b) Each Gᵢ₊₁/Gᵢ is Nilpotent group
(c) Each Gᵢ₊₁/Gᵢ is Simple group
(d) Each Gᵢ₊₁/Gᵢ is Solvable group

A series {e} = N₀ ⊆ N₁ ⊆ N₂ ⊆ ...... ⊆ Nₙ = G of group G is said to be normal series if :
(a) Each Nᵢ is normal subgroup of Nᵢ₊₁
(b) Each Nᵢ is Maximal subgroup of Nᵢ₊₁
(c) Each Nᵢ is abelian subgroup of Nᵢ₊₁
(d) Each Nᵢ is normal subgroup of G

Which of the following is not correct :
(a) Every abelian group is not solvable

(b) Every subgroup of solvable group is solvable
(c) Solvable group G possesses sub-normal series
(d) Every homomorphic image of solvable group is solvable

If N is a normal subgroup of group G then G will be solvable group if and only if :
(a) N is solvable
(b) G/N is solvable
(c) Both N and G/N are solvable
(d) Neither N nor G/N are solvable

Let K be an extension of field F then degree of field extension is :
(a) Number of elements in K.
(b) Number of elements in K
(c) Power of elements of K
(d) Dimensions of Vector space K (F)

If K is finite extension of field F and T is subfield of K containing F then which of the following is correct.
(a) [T : F]/[K : F]
(b) [T : F]/[K : T]
(c) [K : T]/[T : F]
(d) [K : F]/[K : T]

A field F is said to be algebraically closed if :
(a) Every non constant polynomial in F [x] has a root in F
(b) Every irreducible polynomial in F [x] is of degree 1
(c) Every non constant polynomial in F [x] splits in F [x]
(d) All of the above

If all the extensions of field F are separable then F is called :
(a) Algebraically closed field

(b) Simple Field
(c) Galois Field
(d) Perfect Field

If K is normal extension of field F then which of the following correct :
(a) 0 G (K F) = [K : F]
(b) 0 G (K F) ≤ [K : F]
(c) 0 G [K : F] ≥ [K : F]
(d) None of these

If K is Galoss extension of F then K is :
(a) Finite Extension of F
(b) Normal Extension of F
(c) Separable Extension of F
(d) All of the above

Prove that every finite group has atleast one composition series.

Let H is normal subgroup of G then prove G/H is simple group if and only if H is maximal subgroup of G.

OR
Show that every subgroup of solvable group is solvable.

Show that every Nilpotent group is solvable.

OR
Prove that every finite extension of field F is an algebraic extension.

Define splitting field will example.

OR
Define Algebraically closed fields.

Prove that every extension of Q is separable.

OR
Define Galois group and Normal extension of field.

Let G be a finite group with two composition series
G ⊃ H₁ ⊃ H₂ ⊃ .......... ⊃ Hₙ = {e} and
G ⊃ K₁ ⊃ K₂ ⊃ .......... ⊃ Kₘ = {e}
Then show m = n and the two corresponding composition series of quotient groups are abstractly indentical.

OR
Prove that if a cyclic group has exactly one series, then it is a P- group.

Prove that finite P-group is Nilpotent group.

OR
Define solvable group and
Prove that every homomorphic image of solvable group is solvable.

If L is algebraic extension of F. and K is algebraic extension of F then show that L is also algebraic extension of F.

OR
Let K be an extension of F and Let a₁, a₂, ........., aₙ be n elements in K are algebraic over F then show F (a₁, a₂, ........., aₙ) is finite extension of F.

Prove that if field K is algebraically closed then every non constant polynomial in K [x] splits in K [x] and every irreducible polynomial in K [x] is of degree 1.

OR
Show that every field of characteristics zero is perfect.

Let K be a normal extension of field F of characteristics zero.
Show that there exist one-one correspondence between the set of subfields of K which contain F and set of subgroups of G (K. F).

OR
If K is a finite extension of a field F then show that 0 G (K/F) ≤ [K : F].