189-235A: Basic Algebra I Assignment 6

Due: Wednesday, October 27

1. Show that the polynomial x^{3} − 2 is irreducible in Q[x].

2. List all the monic irreducible polynomials of degree 3 in Z_{2}[x].

3. For which odd primes p ≤ 23 is the polynomial x^{2} +1 irreducible in Z_{p}[x]? Can you detect a pattern?

4. Find a polynomial of degree 2 in Z_{6}[x] that has four roots in Z_{6}. Why does this not contradict the theorem shown in class that a polynomial in F [x] of degree d has at most d roots?

5. Show that the ring Z_{3}[x]/(x^{3} + 2x + 1) is a field with 27 elements.

6. Show that two polynomials f(x) and g(x) in R[x] belong to the same congruence class in R[x]/(x^{2}) if and only if f(0) = g(0) and f′(0) = g′(0), where f′(x) is the derivative of f with respect to x.

7. Find the inverse of [x^{2} + x + 1] in the ring Z_{2}[x]/(x^{3} + x + 1).

8. Write down all the powers of [x] in the finite ring Z_{2}[x]/(x^{3} +x+1). What is the smallest j > 1 such that [x]^{j} = 1? Bonus Questions.

9. If p is an odd prime of the form 3 + 4m, show that the polynomial x^{2} = 1 is irreducible in Z_{p}[x], so that Z_{p}[x]/(x^{2} + 1) is a field.

10. If p is an odd prime of the form 1 + 4m, use Wilson’s Theorem to show that a = (2m)! is a root in Z_{p} of the polynomial x^{2} + 1 in Z_{p}[x].

11. If p is a prime of the form 1 + 4m, show that the ring Z_{p}[x]/(x^{2} + 1) is isomorphic to the ring Z_{p} × Z_{p}.