Question

Abstract Algebra: Prove that the polynomial f(X) = X^{4}
+ X + 1 is irreducible on F_{7}[X].

Answer #1

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

Q7) Factorise the polynomial f(x) = x3 −
2x2 + 2x − 1 into irreducible polynomials in
Z5[x], i.e. represent f(x) as a product of irreducible
polynomials in Z5[x]. Demonstrate that the polynomials
you obtained are irreducible.
I think i manged to factorise this polynomial.
I found a factor to be 1 so i divided the polynomial by (x-1) as
its a linear factor.
So i get the form
(x3 − 2x2 + 2x − 1) =
(x2-x+1)*(x-1) which is...

Abstract Algebra: Prove that F5[X]/(X3 + X
+ 1) is a field with 125 elements and then find [3X2 +
2X + 1]−1.

Show the polynomial x4 − x3 −
2x2 + 6x − 4 is irreducible over Q.

Let p be an odd prime. Let f(x) ∈ Q(x) be an irreducible
polynomial of degree p whose Galois group is the dihedral group
D_2p of a regular p-gon. Prove that f (x) has either all real roots
or precisely one real root.

Prove that for every positive integer n, there exists an
irreducible polynomial of degree n in Q[x].

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

Determine whether or not x^2+x+1 is irreducible or not over the
following fields. If the polynomial is reducible, factor it.
a. Z2
b. Z3
c. Z5
d. Z7

Let
a1, a2, ..., an be distinct n (≥ 2) integers. Consider the
polynomial
f(x) = (x−a1)(x−a2)···(x−an)−1 in Q[x]
(1) Prove that if then f(x) = g(x)h(x)
for some g(x), h(x) ∈ Z[x],
g(ai) + h(ai) = 0 for all i = 1, 2, ..., n
(2) Prove that f(x) is irreducible over Q

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

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