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q-Cyclotomic Cosets Factor Roots of Unity over Finite Fields by Vicfred

Last updated: Sun 27 July 2025

tags: mathematics, finite fields, Frobenius, normal bases

Multiplication by q modulo m groups exponents whose roots share a minimal polynomial over F_q. The example is deliberately concrete; it is a test of the statement, not a substitute for it.

Start locally

For \(q=p^r\), the Frobenius map \(F(x)=x^q\) controls extensions of \(\mathbf F_q\). Its orbits determine minimal polynomials, trace, norm, and the Galois group.

$$ C_a=\{a,aq,aq^2,\ldots\}\pmod m $$

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.

$$ m_{\zeta^a,\mathbf F_q}(x)=\prod_{j\in C_a}(x-\zeta^j) $$

Compute before generalising

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

$$ q=2,\ m=7:\qquad\begin{cases}C_0=\{0\},\\C_1=\{1,2,4\},\\C_3=\{3,6,5\}.\end{cases} $$

The global view

The aligned summary deliberately puts the datum and conclusion on different rows. Mathematically, this is the distinction between specifying an object and proving a property of it.

$$ \begin{aligned} \mathsf{D}\;&:\quad C_a=\{a,aq,aq^2,\ldots\}\pmod m,\\[5pt] \mathsf{C}\;&:\quad m_{\zeta^a,\mathbf F_q}(x)=\prod_{j\in C_a}(x-\zeta^j). \end{aligned} $$

Edge conditions

Frobenius is \(\mathbf F_q\)-linear on an extension but not generally linear over a larger coefficient field. Exponents must match the chosen base.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] m_{\zeta^a,\mathbf F_q}(x)=\prod_{j\in C_a}(x-\zeta^j) \end{gathered}} $$

The result is compact enough to reuse without pretending that the caveat has disappeared. The worked line remains the quickest consistency check.

This article was posted on Fri 23 June 2017. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.