Lagmental Vicfred

Primitive Roots Exist for a Very Specific List of Moduli by Vicfred

Last updated: Tue 03 March 2015

tags: mathematics, congruences, multiplicative order, primitive roots

The unit group modulo n is cyclic exactly for n equal to 1, 2, 4, an odd prime power, or twice one. A small computation will anchor the general statement before the abstraction takes over.

The mathematical object

Congruences turn divisibility into arithmetic in \(\mathbf Z/n\mathbf Z\). The unit group \((\mathbf Z/n\mathbf Z)^\times\) controls which residues can be cancelled, inverted, or assigned a multiplicative order.

$$ (\mathbf Z/n\mathbf Z)^\times\ \text{cyclic} $$

I keep the defining relation \(\mathsf D\) above the derived relation \(\mathsf C\). This exposes whether cancellation used \(x\ne0\) and whether the conclusion is canonical.

$$ n\in\{1,2,4,p^k,2p^k\}\qquad(p\ \text{odd prime}) $$

One explicit computation

The computation below is not a second theorem. It is a checksum for the definitions and a place to inspect the difficult LaTeX at full size.

$$ \begin{cases}(\mathbf Z/14\mathbf Z)^\times=\langle3\rangle\cong C_6,\\(\mathbf Z/8\mathbf Z)^\times\cong C_2\times C_2\ \text{is not cyclic}.\end{cases} $$

Why the identity matters

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.

$$ \begin{aligned} \mathsf{D}\;&:\quad (\mathbf Z/n\mathbf Z)^\times\ \text{cyclic},\\[5pt] \mathsf{C}\;&:\quad n\in\{1,2,4,p^k,2p^k\}\qquad(p\ \text{odd prime}). \end{aligned} $$

Where it can fail

Cancellation modulo \(n\) requires a unit. Dividing both sides by a zero divisor is one of the fastest ways to manufacture a false congruence.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] n\in\{1,2,4,p^k,2p^k\}\qquad(p\ \text{odd prime}) \end{gathered}} $$

A symbolic summary is trustworthy only because the example and limitation remain visible beside it. The box compresses the conclusion without hiding its origin.

This article was posted on Mon 04 March 2013. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.