# Ideals in polynomial ring

I am facing this problem:

Prove that there is a bijection between the monic divisors of $$x^n−1$$ in $$F[x]$$ and the ideals of $$F[x]/\left$$.

I tried to find how the ideals in $$F[x]/\left$$ are represented and find the connection to the monic divisors but I did not find any connection.

I think I am missing something very clear. Any suggestions?

• You need to use or prove (use Euclidean division for it) that the ideals are principal. Then, use the definition of quotient to get that the preimage in $F[x]$ of any such ideal is an ideal in $F[x]$ that contains $\langle x^n-1\rangle$. – user647486 Apr 12 at 15:59
• There's nothing special about $x^n-1$ here. – Lord Shark the Unknown Apr 12 at 16:00
• Use the general correspondence between ideals in a ring and its quotient, and contains = divides for prinicipal ideals. – Bill Dubuque Apr 12 at 16:07

There is nothing special about $$x^n-1$$.
The key fact is that the ideals containing $$\langle f(x)\rangle$$ are exactly those generated by the divisors of $$f(x)$$ and monic divisors are canonical representatives of these ideals.
This follows from the ring isomorphisms theorems for $$F[x]/\langle f(x)\rangle$$ and the fact that $$F[x]$$ is a PID.