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Find the smallest positive primitive root modulo each of the following primes:
23.
43.
71.
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:
f(x) = x, m(x) = x^{2} + 1, and p = 5.
f(x) = x, m(x) = x^{2} - x + 1, and p = 5.
f(x) = x - 2, m(x) = x^{2} + 5 x + 1 , and p = 7.
f(x) = x + 1, m(x) = x^{3} - x^{2} + 1, and p = 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.
F_{4} is defined to be the field F_{2}[t]/(t^{2} + t + 1)F_{2}[t].
How many congruence classes are there of polynomials in F_{4}[x] modulo the polynomial x^{3} + x + 1 ?
Find a primitive element for the ring F_{4}[x]/(x^{3} + x + 1)F_{4}[x] of congruence classes.
Explain why the polynomial x^{3} + x + 1 is irreducible over F_{4}.
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.