4. Find the greatest common divisor of the following pairs p(x) and q(x) of polynomials. If d(x) = gcd(p(x), q(x)), and two polynomials a(x) and b(x) such that a(x)p(x)+b(x)q(x) = d(x).
a) p(x) = 7x³ + 6x² − 8x + 4 and q(x) = x³ + x − 2, where p(x), q(x) ∈ Q[x]
5. Find all zeros for the following polynomial
b) 3x³ − 4x² − x + 4 in Z5

8. Which of the following polynomials are irreducible over Q[x] ?
a) x4 − 2x³ + 2x² + x + 4
b) x − 5x³ + 3x − 2
c) 3x⁵ − 4x³ − 6x² + 6
d) 5x⁵ − 6x⁴ − 3x² + 9x − 15

9. Find all of the irreducible polynomials of degrees 2 and 3 in Z2[x].
11. Prove or disprove: There exists a polynomial p(x) in Z6[x] of degree n with more than n distinct zeros.

15. Let f(x) be irreducible. If f(x)|p(x)q(x), prove that either f(x)|p(x) or f(x)|q(x).

24. Let F be a ring and f(x) = a0 + a1x + . . . + anxⁿ be in F[x]. Define f'(x) = a1 + 2a2x + . . . + nanxⁿ¯¹ to be the derivative of f(x).

a) Prove that (f + g)' (x) = f' (x) + g' (x)
Conclude that we can define a homomorphism of abelian groups

Conclude that we can define a homomorphism of abelian groups
D : F[x] → F[x] by (Df(x)) = f' (x).
b) Calculate the kernel of D if charF = 0.
c) Calculate the kernel of D if charF = p.
d) Prove that (fg)' (x) = f' (x)g(x) + f(x)g' (x)

e) Suppose that we can factor a polynomial f(x) ∈ F[x] into linear factors, say
f(x) = a(x − a1)(x − a2)· · ·(x − an)
Prove that f(x) has no repeated factors if and only if f(x) and f' (x) are relatively prime.

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Abstract Algebra Problems
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