Cryptography Problem ! Let F_3 denote the field Z/3Z. Consider the polynomial x^

Question

Cryptography Problem !

Let F_3 denote the field Z/3Z. Consider the polynomial x^2 + 1 epsilon F_3[x]. Prove that x^2 + 1 is irreducible in F_3[x] and that F_3[x]/(x^2 + 1) is a field. List all of the elements field in F_3[x]/(x^2 +1) (you should have 9 elements in all). From now on we will denote this field by F_9. Let F*_9 denote the set of non-zero elements of F_9. For q = p^m a prime power and F_9 a field of q elements, a generator of F*_q is an element a with ord_q(a) = q - 1. Find all the elements in F*_9 (i.e. the non-zero elements in F_3[x]/(x^2 + 1)) which are generators.

Explanation / Answer

a.

The polynomial x2+ 1 is irreducible over F3 since it has no roots in F3.

For a prime p and a monic irreducible f(x) in Fp [x] of degree n , the ring F p[x]/f(x) is a field of order pn.

Proof.

The cosets mod f(x) are represented by remainders

c0+c1x+......cn-1x^(n-1)

and there are

p^n of these.. Since the modulus f(x) is irreducible,, the ring F p[x]/f(x) is a field using the same proof that Z/(m) is a fieeld when m is prime.

b)

9 elements are-a+bx where a,b are {0,1,2}.

c)x+! is a generator. group cyclic of order 8. check which element generates entire F

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