Ideals in polynomial ring

Question

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 $F[x]/\left<x^n−1\right>$.

I tried to find how the ideals in $F[x]/\left<x^n−1\right>$ 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 '19 at 15:59
Use the general correspondence between ideals in a ring and its quotient, and contains = divides for prinicipal ideals. — Bill Dubuque, Apr 12 '19 at 16:07

score 0 · Answer 1 · answered Apr 12 '19 at 16:02

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.

Ideals in polynomial ring

1 Answers1