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Let $F$ be a field. Let $p$ and $q$ be monic members $F[x]$. Let $I = \{p\cdot r : r\in F[x]\}$ and $J = \{q\cdot r : r\in F[x]\}$. I know that if $F[x]/I$ is isomorphic to $F[x]/J$ then ($\operatorname{deg}(p) = \operatorname{deg}(q)$ and (p is reducible if and only if q is reducible)). For cases where we do have $\operatorname{deg}(p) = \operatorname{deg}(q)$ and (p is reducible if and only if q is reducible), is there any faster way to check whether $F[x]/I$ is isomorphic to $F[x]/J$ than seeing if any element of $F[x]/I$ has q as its minimal polynomial?

(In the particular cases I'm interested in, $F$ is very finite.)

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For finite fields, factorization is what you need. – Hurkyl May 16 '11 at 5:19
up vote 7 down vote accepted

If $F$ is finite, $p$ and $q$ are irreducible, and have the same dagree $d$, then $F[x]/I$ and $F[x]/J$ are isomorphic $F$-algebras, since there is (up to isomorphism) only one extension field of a finite field of any given degree (see for example Birkhoff & Mac Lane, A survey of Modern Algebra, Section 15.6).

Things get a little more interesting when $p$ and $q$ are not irreducible. Suppose that $p=p_1^{n_1}p_2^{n_2}\ldots$. Then $F[x]/I$ is isomorphic to $\bigoplus_i F[x]/(p_i^{n_i})$. It turns out that, for any perfect field $F$, if $E_i=F[x]/(p_i)$, then $F[x]/(p_i^{n_i})$ is isomorphic to $E_i[t]/(t^{n_i})$ (this result is well-known, if not well-documented; a proof appears here (see Theorem A.19)). Thus for finite fields the general answer to you question is the following:

For finite fields

$F[x]/I$ and $F[x]/J$ are isomorphic if and only if the numerical invariants of the polynomials $p$ and $q$ are the same, in other words, if $p=p_1^{n_1}p_2^{n_2}\ldots p_r^{n_r}$ and $q=q_1^{m_1}q_2^{m_2}\ldots q_s^{m_s}$ are the decompositions of $p$ and $q$ into irreducible factors, then there is a bijection $w:\{1,\ldots,r\}\to \{1,\ldots,s\}$ such that the degree of $q_{w(i)}$ is the same as the degree of $p_i$ and $m_{w(i)}=n_i$ for all $i\in \{1,\ldots,r\}$.

For general perfect fields, this does not give a complete solution. What we get is

For perfect fields

$F[x]/I$ and $F[x]/J$ are isomorphic if and only if when$p=p_1^{n_1}p_2^{n_2}\ldots p_r^{n_r}$ and $q=q_1^{m_1}q_2^{m_2}\ldots q_s^{m_s}$ are the decompositions of $p$ and $q$ into irreducible factors, then there is a bijection $w:\{1,\ldots,r\}\to \{1,\ldots,s\}$ such that $F[x]/(q_{w(i)})$ is isomorphic to $F[x]/(p_i)$ and $m_{w(i)}=n_i$ for each $i\in \{1,\ldots,r\}$.

Thus the problem is reduced to determining whether or not two irreducible polynomials give rise to isomorphic field extensions.

Edit: Ricky Demer pointed out that the "only if" parts of the above assertions are not obvious, and so here is a way out: We can recover the subalgebras $F[x]/(p_i^{m_i})$ from $F[x]/I$ as the indecomposable ideals. It remains to show that if $p_i$ is irreducible then the field $E_i=F[x]/p_i$ and the integer $m_i$ can be recovered from $F[x]/(p_i^m)$. This is not hard:

  1. The field $F[x]/p_i$ can be recovered from $F[x]/(p_i^n)$ as the quotient of the algebra by its radical.

  2. The integer $m_i$ can be recovered as the length of the ring $F[x]/p_i^{m_i}$.

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How do you get "and only if"? – Ricky Demer May 16 '11 at 5:48
@Ricky: for finite fields I think you can do it by counting the number of roots of various polynomials and induction. Alternately, you can apply the structure theorem to modules over $F[x]$ (since if $\phi : F[x]/I \to F[x]/J$ is an isomorphism of rings you get an isomorphism of $F[x]$-modules by letting $x$ act as $\phi(x)$ on $F[x]/J$.) – Qiaochu Yuan May 16 '11 at 5:52
Oh, you also need to transport $F$, I guess. – Qiaochu Yuan May 16 '11 at 5:52
re edit: We still need to show that the factors ($E_i$ and $n_i$) can be recovered from $F[x]/I$. – Ricky Demer May 16 '11 at 6:49
The fact that (for $p(x) \in F[x]$ irreducible) $F[x]/p^n$ is isomorphic to $E[t]/t^n, E = F[x]/p$ is (essentially equivalent to and) a consequence of the fact that the completion of $F(x)$ at the place $p$ is isomorphic to the power series field $E((t))$. This is the well-known characterization of equicharacteristic complete discretely valued fields. This you can find in most books on local fields, e.g., Serre's. – Felipe Voloch May 16 '11 at 11:13

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