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 $q(x)$


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?

  • $\begingroup$ 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
  • 4
    $\begingroup$ They are exactly the same. $\endgroup$ – M. Van Sep 17 '17 at 16:57
  • 3
    $\begingroup$ 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
  • 3
    $\begingroup$ @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
  • 2
    $\begingroup$ @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

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.