Lagmental Vicfred

Factorisation Modulo p Reveals Cycle Types by Vicfred

tags: mathematics, polynomials, discriminants, Galois groups

At good primes, the degrees of irreducible factors give the cycle lengths of a Frobenius element. I want the notation, the mechanism, and the failure mode visible at the same time.

Set-up

For a separable polynomial \(f\in K[x]\), the Galois group permutes its roots faithfully. Factorisations, discriminants, and resolvents constrain the resulting subgroup of \(S_n\).

$$ f\bmod p=f_1\cdots f_r,\qquad\deg f_i=d_i $$

There are two layers here: the object \(\mathsf D\) and the law \(\mathsf C\). Writing them separately makes the direction of \(\Longrightarrow\) visible and keeps an accidental converse from slipping in.

$$ \operatorname{Frob}_p\ \text{has cycle type }(d_1,\ldots,d_r) $$

The calculation

The middle display is intentionally dense: it is where signs, bounds, multiplicities, or normalising factors are most likely to be lost.

$$ \begin{array}{c|c|c}f\bmod p&\text{factor degrees}&\text{cycle type}\\\hline(1)(3)&1,3&(1)(3)\\(2)(2)&2,2&(2)(2)\end{array} $$

What survives abstraction

The compact alignment is a local map of the argument: assumptions on the first row, consequence on the second. Any generalisation must preserve that dependency.

$$ \begin{aligned} \mathsf{D}\;&:\quad f\bmod p=f_1\cdots f_r,\qquad\deg f_i=d_i,\\[5pt] \mathsf{C}\;&:\quad \operatorname{Frob}_p\ \text{has cycle type }(d_1,\ldots,d_r). \end{aligned} $$

The boundary

A discriminant square distinguishes containment in \(A_n\), but it usually does not determine the entire Galois group by itself.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \operatorname{Frob}_p\ \text{has cycle type }(d_1,\ldots,d_r) \end{gathered}} $$

This is enough machinery for one note: an exact object, a worked case, a structural law, and a clearly marked boundary. Each layer can now be tested independently.

This article was posted on Wed 01 October 2025. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.