# When will $R_q=\mathbb{Z}[X]/\langle X^n+1\rangle$ be a field? [closed]

When will $$R_q=\mathbb{Z}[X]/\langle X^n+1\rangle$$ be a field?

• It's asked the conditions such that it's obtained a field by taking the set of residue classes of polynomials of variable $X$ raised to non-negative powers with coefficients in $\mathbb Z$, per the equivalence relation "has the same remainder by polynomial division by the polynomial $X^n+1$". This is a mathematical question with direct relevance to cryptography. That should be covered in any course with followup question asking that. If such course is not available, see section 2.5.4 of the Handbook of Applied Cryptography
– fgrieu
Sep 18 at 6:43
• If $R$ is a ring then $R/I$ is a field iff if $I$ is a maximal ideal of $R$. Sep 18 at 18:54
• The notation $R_q$ typically means $(\mathbb{Z}/q\mathbb{Z})[x] / \langle x^n+1\rangle$,but your $R_q$ does not appear to depend on $q$.
– Mark
2 days ago
• Nevermind - my comment was wrong
– tylo
2 days ago