I know how to find the order of a finite field. But how do you do it with a finite field modulo a polynomial $m(x)$ ? For example how do you find the order of $\mathbb{Z}_3[x]_{x^3+2x^2+1}$ ?
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It's a consequence of division with remainder that $K[x]/p$ is a $K$-vector space of dimension $\deg p$ (or, in case $p=0$, of dimension $\aleph_0$). Whether or not it's a field, it depends on $p$. – Sassatelli Giulio Nov 28 '24 at 13:24
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$x^2+x+1$ is not prime, so this won't be a field, but the order will be, as a ring, $p^{\deg(m)},$ in general – Thomas Andrews Nov 28 '24 at 13:26
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@ThomasAndrews: OP has $x^3+2x^2+1$ – J. W. Tanner Nov 28 '24 at 13:30
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Even better than my previous link: this question is an exact duplicate of Polynomials, finite fields and cardinality/dimension considerations (found using the search engine of MSE: https://math.stackexchange.com/search?q=dimension+and+cardinality). – Anne Bauval Nov 28 '24 at 13:36
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Hint: your polynomial is irreducible (plug in $0,1$ and $2$, they are not a zero.) Hence the quotient is a field and isomorphic to $\mathbb{F}_{27}$. – Nicky Hekster Nov 28 '24 at 13:45
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1Sure, after an edit. @tanner You're here long enough to know how this works. – Thomas Andrews Nov 28 '24 at 14:44