Cardinality of the ring $F_3[x] / (x^2-x+1)$

Question

How would I find the number of elements of the ring $F_3[x] / (x^2-x+1)$?

I know that $x^2-x+1$ is not prime/irreducible, since gcd($x^2-x+1$, $x^3-x^2-1$) = 3.

Can anyone provide some tips?

Hint: see the second paragraph of this answer. – Bill Dubuque Feb 14 '15 at 21:27 — Bill Dubuque, Feb 14 '15 at 21:27

score 1 · Accepted Answer · answered Feb 14 '15 at 21:34

1

It is not relevant if the polynomial is irreducible or not to find the number of elements of the ring.

Given a non-constant polynomial $g$ and a field $F$, we have $F[X]/(g)$ is an $F$-vectorspace of dimension $\deg g$.

As such its cardinality is $|F|^{\deg g}$.

(If you want to say something on the multiplicative structure of the ring, then the factorization gets important.)

answered Feb 14 '15 at 21:34

quid

The degree of g is 2 right? What is |F|? – jstnchng Feb 14 '15 at 21:38
Yes the degree is $2$ and $|F|$ is the cardinality of your field $F_3$, which should be $3$. – quid Feb 14 '15 at 21:42
1

Ah I see!! Thanks quid. – jstnchng Feb 14 '15 at 21:42

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