Motivation for cyclotomic units

Question

I am wondering the original motivation for considering cyclotomic units. Maybe one can rephrase the question as:

1. Why did people initially consider such units in $\mathbb{Q}(\zeta_p)$ specially?

There are many interesting results we can get with the notion of cyclotomic units, for example, comparison between the class number and the index of the subgroup of cyclotomic units. However, I think that they are not the philosophical origin of cyclotomic units. One another but stupid rephrase of the question is:

2. Why was the name cyclotomic taken only for such units, not for every unit in $\mathbb{Q}(\zeta_p)$?

Answer 1

I'm quite sure cyclotomic units were first introduced by Kummer in his 1847 paper, where he proved his very famous partial solution to Fermat's Last Theorem:

Theorem (Kummer) If $p$ is an odd prime that does not divide the class number of the field $\mathbb{Q}(\zeta_p)$ and $(xyz,p)=1$, then

$$x^p+y^p=z^p$$

has no rational solutions.

In order to get the proof started, he needed to study $\mathbb{Z}[\zeta_p]$, and in particular, certain units, which we know call cyclotomic.

As for the name, I guess they wanted to single out the particularly relevant ones (and only those).

The first chapter in Lawrence Washington's "Introduction to Cyclotomic Fields" is a great exposition of the proof.