Classical Algebra (Math 326) Written Assignment

due Monday, May 7, 2001

Directions

There will be a premium placed on accuracy in the grading of this assignment. Please submit your assignment typed. If there is more than one page, please staple. Explain your solutions.

Problems

1. Find primitive roots modulo the following primes:

(a) 23.

(b) 53.

(c) 71.

2. Find the order of the congruence class of the polynomial f(x) modulo the polynomial m(x) when the field of coefficents is F_{p} in the following cases:

(a) f(x) = x, m(x) = x^{2} + 1 , and p = 5 .

(b) f(x) = x, m(x) = x^{2} − x + 1 , and p = 5 .

(c) f(x) = x − 2 , m(x) = x^{2} + 5x + 1 , and p = 7 .

(d) f(x) = x + 1 , m(x) = x^{3} − x^{2} + 1 , and p = 3 .

3. Show that over any field F the polynomial x and the polynomial f(x) are coprime polynomials if and only if f(0) ≠ 0.

4. Find a polynomial f(t) in F_{5}[t] whose congruence class modulo m(t) is a primitive element for the field F_{5}[t]/m(t)F_{5}[t] when m(t) = t^{2} − t + 1.

5. F_{4} is defined to be the field F_{2}[t]/(t^{2} + t + 1)F_{2}[t].

(a) Explain why the polynomial x^{3} + x + 1 is irreducible over F_{4}.

(b) Find a primitive element for the field F_{4}[x]/(x^{3} + x + 1)F_{4}[x].