MAT 3343, APPLIED ALGEBRA, FALL 2003 Problem Set 5, due November 21, 2003 Peter Selinger

Problem 1. How many roots does the polynomial p(x) = x^{2} + x + 8 have in ℤ_{10}? Why does this not contradict the root theorem? Problem 2. (a) Let F be a field, and let p(x) = a_{3}x^{3} + a_{2}x^{2} + a_{1}x + a_{0} be a third-degree polynomial in F [x],

with a_{3} ≠ 0. Prove that p(x) is irreducible if any only if p(x) has no roots.

(b) Show that the analogous statement is not true for fourth-degree polynomials, i.e., give an example of a field F and a fourth-degree polynomial q(x) = a_{4}x^{4} + a_{3}x^{3} + a_{2}x^{2} + a_{1}x + a_{0} such that q(x) is reducible but has no roots.

Problem 3. Find the gcd d(x) of p(x) = x^{8} + x^{7} + x^{5} + x^{3} + x^{2} + x + 1 and q(x) = x^{7} + x^{6} + x^{4} + x^{3} + 1 in ℤ_{2}[x]. Find s(x) and t(x) such that p(x)s(x) + q(x)t(x) = d(x). Problem 4. (a) Find all irreducible polynomials of degree up to 5 in ℤ_{2}[x].

(b) Factor p(x) = x^{12} + x^{10} + x^{7} + x^{6} + 1 into irreducible factor in ℤ_{2}[x]. Problem 5. Factor the following polynomials into irreducible factors in the given ring. In each case, give a reason why you know that the factors you found are irreducible.

(a) x^{5} − 1 in ℚ[x].

(b) x^{5} + 1 in ℤ_{2}[x].

(c) x^{4} + 1 in ℤ_{5}[x].

(d) 2x^{3} + x^{2} + 4x + 2 in ℚ[x].

(e) x^{4} − 9x + 3 in ℚ[x].

(f) x^{8} − 16 in ℚ[x]. Problem 6. Find all irreducible polynomials of the form x^{2} + ax + b over ℤ_{5} (i.e., all irreducible polynomials of degree 2, with leading coefficient 1). Problem 7. Are the following polynomials irreducible in the given ring? Give reasons.

(a) x^{3} + x^{2} + x + 1 in ℚ[x].

(b) 3x^{8} − 4x^{6} + 8x^{5} − 10x + 6 in ℚ[x].

(c) x^{4} + x^{2} − 6 in ℚ[x].

(d) 4x^{3} + 3x^{2} + x + 1 in ℤ_{5}[x]. Problem 8. In each case, find a polynomial in ℚ[x] with a as a root. Then prove that a is irrational.

(a) a = \sqrt{2}/\sqrt[3]{5}.

(b) a = \sqrt{2} + \sqrt{3}. Problem 9. Find all rational roots of 3x^{3} + 4x^{2} − x − 2.