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Let $F_p[x]$ be a ring of polynomials with coefficients in the field $F_p$. Let $q(x)$ be some irreducible polinomial in the base field $F_p$ with degree $n$. To construct the field extension $F_p^n$, I've seen in literature two different notations/constructions:

1) $F_p^n = F_p[x]/\langle q(x) \rangle$ : quotient ring by the ideal generated by $q(x)$

and

2) $F_p^n = F_p[x]/( q(x) )$ : the residue class by $q(x)$, when operations are done $\mod q(x)$.

Now I ask: These different notations means the same? Why one might be preferable instead the other?

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  • $\begingroup$ See this answer for the general relationship between congruences, quotient rings and subalgebras of the square. $\endgroup$ – Bill Dubuque Sep 17 '17 at 16:56
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    $\begingroup$ They are exactly the same. $\endgroup$ – M. Van Sep 17 '17 at 16:57
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    $\begingroup$ quotienting out an ideal $I$ and doing arithmetic with the cosets is exactly what you get when you quotient out by the equivalence relation $a ~ b$ iff $a-b \in I$ and define addition and multiplication on the equivalence classes in the natural way. $\endgroup$ – M. Van Sep 17 '17 at 16:59
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    $\begingroup$ @BillDubuque How are they not the same sets with the same ring structure on them (not only isomorphic, but the isomorphism is given by the identity -.-)? $\endgroup$ – M. Van Sep 17 '17 at 17:15
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    $\begingroup$ @Bill Dubuque, is it not true that by definition $R/I = R/\!\equiv$ as sets, where $I$ is ideal in $R$ and $\equiv$ is congruence generated by $I$? $\endgroup$ – Ennar Sep 17 '17 at 21:46
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These are exactly the same (note though that this field is written $\mathbb{F}_{p^n}$, not $\mathbb{F}_p^n$). "Residue classes mod $q(x)$" and "elements of the quotient by the ideal generated by $q(x)$" are two different terms for the same thing: namely, equivalence classes of elements of $\mathbb{F}_p[x]$ under the equivalence relation $f(x)\sim g(x)$ iff $f(x)-g(x)$ is a multiple of $q(x)$. The difference in notation between $(q(x))$ and $\langle q(x)\rangle$ is meaningless--some people use parentheses to denote ideals generated by a list of elements, and other people use angle brackets. You should feel free to use either one, as long as you are consistent.

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