Trying to get through a comprehension sticking point in arithmetic in a Galois field.
$GF(8)$ has the following irreducible polynomials: (https://people.computing.clemson.edu/~westall/851/rs-code.pdf)
$x{^3}+x+1$ and $x{^3}+x{^2}+1$
In the above reference, $5 \cdot 6$ yields $(x+1)$ remainder $(x+1)$ using the first polynomial.
When I multiply using the second polynomial, I obtain $(1)$ remainder $(x{^2})$
My question is, am I doing the arithmetic wrong, or are arithmetic results in a finite field dependent upon the irreducible polynomial (i.e., a polynomial defines a "unique" field)?