Adv Abstract Algebra-I

Sub normal series (e) = G₀ ⊆ G₁ ⊆ G₂ ⊆ ............G_n = G of group G is called composition series of G of each of its factor group G_i/G_i+1 is

(a) Simple group

(b) Abelion group

(c) Cyclic group

(d) Solvable group

Any two composition series of a finite group are

(a) Abstractly , identical

(b) , Having equal number of members

(c) Both the above are correct

(d) None is correct

Let N be a normal subgroup of any group G then G is solvable group if.

(a) N is solvable

(b) G/N is solvable

(c) Both N and G/N solvable

(d) None of above

Every abelian group is

(a) Simple group

(b) Cyclic group

(c) P-group

(d) Solvable group

If an extension field E of F is not algebraic extension it is called

(a) Normal extension

(b) Simple extension

(c) Separable extension

(d) Transcendental extension

If L is finite extension of K, and K is finite extension of F, then which of the following is true

(a) [L : K] / [L : F]

(b) [L : K] [K : F]

(c) [L : F] / [L : K]

(d) [K : F] / [L : K]

If field F is said to be perfect field if all the extension of field F are.

(a) Algebraic extension

(b) Normal extension

(c) Separable extension

(d) Simple extension

If K is algebraically closed field then every irreducible polynomial in K [x] is of degree.

(a) 0

(b) 1

(c) finite

(d) Infinite

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

If T is any subfield of K which contain F then [T:F]=

Define composition series.

OR

A normal subgroup H of G is maximal if and only if G/H is simple.

Define solvable and Nilpotent group.

OR

Show that every group of order Pⁿ is Nilpotent. Here P is prime.

Define algebraic extension and separable extension.

OR

Let a ∈ k is algebraic element over field F the show that any two minimal monic polynomial for 'a' are equal.

Define algebraically closed fields.

OR

Show that every field of characteristics zero is perfect.

Define outomorphism of extension and show that Fixed field of G is subfield of K. Where G is subgroup of all automor-phism of field k.

OR

Define galois extension and also prove that G (K,F) is a subgroup of the group of all automorphism of K.

Show that if G is a commutative group having a composition series then G is finite.

OR

State and prove Jordon - Holder theorem.

Prove that subgroup of a solvable group is solvable.

OR

Let G be a nilpotent group. Then show that every subgroup of G and every homomorphic image of G are nilpotent.

Let K be an extension of field F then prove that an element a ∈ k is algebraic over F if and only if F (a) is finite extension of F.

Define perfect field. Also show that if E is a finite separable extension of field F then E is simple extension of F.

OR

Prove that for any field K that K is algebraically closed if and only if every polynomial in K (x) of positive degree has at least one root in K.

Let E be a Galois extension of F let K be any subfield of E contain F then show that there exist one-one correspondence from the set of subfield of E contain F to the subgroup of G (E/F) such that G [E/F] = [E:K].

OR

Prove that the general polynomial of degree n ≥ 5 is not solvable by radicals.