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Let $F$ be a field. Let $F[x]$ be the polynomial ring and $f(x)$ and $g(x)$ be 2 irreducible polynomials of same degree in $F[x]$. When do we have the 2 fields $$F[x]/<f(x)> \cong F[x]/<g(x)>$$

Well, I can think of $F[x]/<f(x)>$ as the field $F(\alpha)$ where $\alpha$ is the root of $f(x)$ and check if any algebraic combination of $\alpha$ with elements of $F$ is a root of $g(x)$. If it is then both the fields are the same/isomorphic. But is there any general condition on the 2 polynomials such that the isomorphism holds?

Jagdeep Singh
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  • Small comment (which might not even be totally correct), but I think in the quadratic case $f(x) = ax^2+bx+c$, the field $F[x]/(f)$ obtained only depends up to isomorphism on the discriminant, $\Delta = b^2-4ac$ (roughly because once $\sqrt{\Delta}$ is adjoined, you have everything). – Mike F Nov 06 '16 at 05:27
  • If $F$ is a finite field then it is true for any polynomials ! So, No $F$ is not a finite field. – Jagdeep Singh Nov 06 '16 at 05:38
  • Strongly related: https://math.stackexchange.com/questions/1831305 – Watson Nov 06 '16 at 09:27

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