Irreducible Polynomial over a Field

Let K be obtained as a field Q(alpha) where alpha is a root of P(x) = x3 −3. Find an irreducible polynomial which defines the splitting field of P(x).

If P is a prime number, prove that the polynomial x^n - p is irreducible over the rationals.

Modern Algebra Ring Theory (XXXII) Polynomial Ring Irreducible Polynomial If f(x) is in F[x], where F is the field

Modern Algebra Ring Theory (XXXI) Polynomial Ring Irreducible Polynomial Let F be a field of real nu

Prove that x^2 + 1 is irreducible over the field F of integers mod 11 and prove directly that F[x]/(x^2 + 1) is a field having 121 elements.

Prove that the polynomial x^2 + x + 1 is irreducible over F , the field of integers modulo 2. The complete problem is in the attached file.

Problem #4 A palindromic polynomial is such that for all . Now we use the result in problem #3 to find an irreducible palindromic polynomial of degree 6 over . First, we compute all reducible ones. Now we consider the palindromic . From problem #3(c), is irreducible. I am sorry but I do not understand the definit

Find an irreducible polynomial defining the extension Q(3^1/2, 5^1/2).

1) How do I show x+1 is primitive? 2) How do I prove x^4+x^3+1 is an irreducible polynomial of degree 4 over Z mod?