Question

Consider the polynomial (x) = x³ − x + 1 ∈ ℤ₃[x] . Since (x) has no roots in ℤ₃ and it has degree 3 it is irreducible. Thus the ring :=ℤ₃[x]/((x)) is a finite field of order 27. Let ∈ denote the element x+((x)) , so that ()=0 .

It is proven that ₓ is cyclic. Find (with proof!) an element r∈ₓ which is a generator. In other words, find an element r such that every element in ₓ is of the form rₙ for an integer n

Answer 1

To find a generator in the field ₓ, we can define r as the residue class of x in the quotient ring. We need to show that every element in ₓ can be expressed as a power of r. To determine if r is a generator, we can raise it to the powers 0, 1, 2, ..., 26 and check if we obtain all distinct elements in ₓ.

Step-by-step explanation:

To find an element r that is a generator in the field ₓ, we can use the fact that the field ₓ is cyclic. Since ₓ is a finite field of order 27, the generator r must have a multiplicative order of 27.

We know that ₓ = ℤ₃[x]/((x)), and we can define r as the element x + ((x)). In other words, r is the residue class of x in the quotient ring.

To prove that r is a generator, we need to show that every element in ₓ can be expressed as a power of r. Since ₓ has order 27, we can start by raising r to the powers 0, 1, 2, ..., 26 and check if we obtain all distinct elements in ₓ. If all distinct elements are obtained, then r is a generator of ₓ.