Classical Algebra (Math 326) Written Assignment

due Monday, Dec 9, 2002

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 the smallest positive primitive root modulo each of the following primes:

(a) 23.

(b) 43.

(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. 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.

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

(a) How many congruence classes are there of polynomials in F_{4}[x] modulo the polynomial x^{3} + x + 1?

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

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

5. Write a proof of the following proposition: If F is a field and f(x) is in the ring F [x] of polynomials with coefficients in F, then the polynomial x and the polynomial f(x) have no (non-constant) common factor if and only if f(0) ≠ 0.