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?

  • $\begingroup$ 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 $\langle x^n-1\rangle$. $\endgroup$ – user647486 Apr 12 at 15:59
  • $\begingroup$ There's nothing special about $x^n-1$ here. $\endgroup$ – Lord Shark the Unknown Apr 12 at 16:00
  • $\begingroup$ Use the general correspondence between ideals in a ring and its quotient, and contains = divides for prinicipal ideals. $\endgroup$ – 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.


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