Question

Abstract Algebra: Prove that F_{5}[X]/(X^{3} + X
+ 1) is a field with 125 elements and then find [3X^{2} +
2X + 1]^{−1}.

Answer #1

Abstract Algebra : Find the splitting field of the polynomial
f(X) = X3 + 2pX + p in Q[X]. P is a prime number.

Abstract Algebra: Prove that the polynomial f(X) = X4
+ X + 1 is irreducible on F7[X].

Abstract Algebra (Modern Algebra)
Prove that every subgroup of an abelian group is abelian.

*GROUP THEORY/ABSTRACT ALGEBRA*
If a ∈ G and a^m = e, prove that o(a) | m

Abstract algebra
Either prove or disprove the following statements. Be sure to
state needed theorems or supporting arguments.
Let F ⊂ G be finite fields. Then F, G have the same
characteristic,say p. Moreover, if p > 0 then logp (|F|) divides
logp (|G|).

(abstract alg) Let G be a cyclic group with more than two
elements:
a) Prove that G has at least two different generators.
b) If G is finite, prove that G has an even number of
generators

Abstract Algebra, Using the knowledge of Unique Factorization
Domains to solve:
Let D be an integral domain and D[x] the polynomial ring over D.
Show that if every nonzero prime ideal of D[x] is a maximal ideal,
then D is a field.

Evaluate the following integral. ∫ (x3 − 3x2)( 1 x −
3) dx
(A) − 3 4 x4 + 14 3 x3 − 3 2 x2 + C
(B) − 3 4 x4 + 4 x3 − 3x2 + C
(C) − 3 4 x4 + 10 3 x3 − 3x2 + C
(D) − 3 4 x4 + 10 3 x3 − 3 2 x2 + C
(E) ( 1 4 x4 − x3)( 1 x − 3) + ...

(a) Let a,b,c be elements of a field F. Prove that if a not= 0,
then the equation ax+b=c has a unique solution.
(b) If R is a commutative ring and x1,x2,...,xn are independent
variables over R, prove that R[x σ(1),x σ (2),...,x σ (n)] is
isomorphic to R[x1,x2,...,xn] for any permutation σ of the set
{1,2,...,n}

g(x)=[x3(3x-1)^2(2x+1)]^1/2
find the derivative

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