There are many important applications of units in algebraic number theory in general and in cyclotomic rings of integers in particular. We have the analytic class number formula for the field K
$$\lim_{s\to1} (s-1)\prod_{I}\frac{1}{1-N^s(I)}=\frac{2^{r_1+r_2}\pi^{r_2}R_{K}}{m\sqrt{\vert D_{K}\vert}}h_K$$
with a product that ranges over all prime ideals I of the algebraic ring of integers on the left hand side. On the right hand side there are the $r_1$ real embeddings and the $r_2$ complex embeddings of the number field K, but we also need to know the regulator $R_K$ of the fundamental system of units, the number m of roots of unity and the discriminant $D_K$. The formula then yields the class number $h_K$. If we found the regulator $R_K$ there are good chances that we have the discriminant $D_K$ and the remaining values on the right hand side too. The computation of the prime ideals on the right hand side is often possible (see here for example) so that you get a relatively good approximation for the class number $h_K$. If you ever want to determine a class group you will learn that it is very helpful to know the class number.
A very famous use of the units of cyclotomic rings of integers is connected to Fermat's last theorem. Kummer needed to know whether the class number of the $p^{\text{th}}$ cyclotomic ring of integers, p an odd prime, is divisible by the prime $p$ and whether a unit that is congruent to a (rational) integer modulo $p$ is the $p^{\text{th}}$ power of another unit. For so called regular primes he confirmed it in the positive. And he found out as follows:
He could compute the first factor $h_1$ of the class number $h$. But the second factor $h_2$ is the class number of the real cyclotomic field of integers or the index of the quotient module of the fundamental system of units of the cyclotomic ring of integers modulo the module spanned by the units
$$\tag{1}\left[\pm 1,\zeta, \frac{1-\zeta^2}{1-\zeta}, \frac{1-\zeta^3}{1-\zeta},\dots,, \frac{1-\zeta^\mu}{1-\zeta}\right],\mu=\frac{p-1}{2}.$$
If the index is divisible by the prime $p$ there must exist a unit
$$\varepsilon_i:=b_0+b_1\zeta+b_2\zeta^2+\dots+b_{p-1}\zeta^{p-1}$$
of order $p$ in the quotient module by elementary group theory and we have
$$\varepsilon_i^p\equiv c \pmod{p}$$
because $(\zeta^k)^p=1$ and all binomials $\binom{p}{j}$, $0<j<p$, are divisible by the prime $p$. But the unit $\varepsilon_i^p$ is an element of $(1)$ thus we have
$$\tag{2}\zeta^k\prod_{j=2}^\mu\left(\frac{1-\zeta^j}{1-\zeta}\right)^{x_j}=\varepsilon_i^p\equiv c \pmod{p}.$$
Not all exponents $x_j$ can be divisible by the prime $p$: otherwise the unit $\epsilon_i$ has order $1$ in the quotient module. Kummer replaced the $p^{\text{th}}$ root of unity $\zeta$ by the variable X in this congruence and multiplied out the polynomials on the left hand side with
$$q_j(X):=\frac{1-X^j}{1-X}=1+X+X^2+\dots+X^{j-1}$$
and obtained
$$\tag{3}X^k\prod_{j=2}^\mu\left(\frac{1-X^j}{1-X}\right)^{x_j}= c+p\Omega(X)+(1+X+X^2+\dots+X^{p-1})\Psi(X)$$
by the assumption $(2)$ with some polynomials $\Omega(X)$ and $\Psi(X)$. The term
$$m(X):=1+X+X^2+\dots+X^{p-1}$$
is needed because we are working in the ring $\mathbb Z[\zeta]/m(\zeta)$ so that two cyclotomic integers are equal if they are congruent modulo $m(\zeta)$. Now Kummer took the logarithmic derivative of $(3)$ or he differentiated with respect to X and divided through the unit:
\begin{align}
\tag{4}\frac{k}{X}&+\sum_{j=2}^\mu x_j \frac{q_j'(X)}{q_j(X)} \\
&=\frac{p\Omega '(X)+(1+2X+3X^2+\dots+(p-1)X^{p-2})\Psi(X)+m(X)\Psi '(X)}{c+p\Omega(X)+(1+X+X^2+\dots+X^{p-1})\Psi(X)}.
\end{align}
In plugging in $X=\zeta$ and working modulo p Kummer could prove that such a unit $(2)$ with at least one exponent $x_j$ not divisible by the prime $p$ does not exist if the numerators of the Bernoulli numbers $B_2$, $B_4$, $\dots$, $B_{p-3}$ are not divisible by the prime $p$. Edwards proves this result in chapter $6.17$ of his book Fermat's last theorem. With this he easily proves in the next chapter that a unit that is congruent to an integer modulo p is a $p^{\text{th}}$ power of (another) unit.
Now we are following another interesting line. We multiply equation $(4)$ with $X(1-X)$ and obtain first
$$X(1-X)\frac{q_j'(X)}{q_j(X)}=
{X(1-X)\frac{1-X}{1-X^j}\frac{d\left(\frac{1-X^j}{1-X}\right)}{dx}}=
{X(1-X)\frac{1-X}{1-X^j}\cdot\frac{-j(1-X)X^{j-1}+(1-X^j)}{{(1-X)^2}}}=
{-jX^j\frac{1-X}{1-X^j}+X}.$$
But the cyclotomic integer $q_j(\zeta)$ is a unit so that the inverse exists. With $\hat j j\equiv 1\pmod{p}$, $0<\hat j<p$ and the homomorphism $\sigma_j: \zeta\to \zeta^j$ we obtain
$$\frac{1-\zeta}{1-\zeta^j}={\sigma_j\left(\frac{1-\zeta^{\hat j}}{1-\zeta}\right)}=
{\sigma_j\left(1+\zeta+\zeta^2+\dots+\zeta^{\hat j-1}\right)}=
{1+\zeta^j+\zeta^{2j}+\dots+\zeta^{j(\hat j-1)}}=:\pi_j(\zeta).$$
The polynomial $\pi_j(\zeta)$ is defined according to the exponents of the second last term and they are computed to a positive number less than $p$ with $\zeta^p=1$. Thus the left hand side of $(4)$ can be written by polynomials in $X$ of maximal degree $p-1$ after the multiplication with $X(1-X)$. The analysis of the right hand side shows that it is congruent to $0$ modulo $\langle p,m(X) \rangle $. Writing the polynomials as vectors over the field $\mathbb F_p$ we can search for combinations of exponents $(x_j)_j$ and $k$ such that
$$\tag{5}k(1-X)+\sum_{j=2}^\mu x_j \left\{-jX^j\pi_j(X)+X\right\}\equiv 0\mod \langle p,m(X) \rangle .$$
The ansatz given here can be improved in taking the units
$$\hat\epsilon_j:=\zeta^{k_j}\frac{1-\zeta^j}{1-\zeta}.$$
with an integer $k_j$ such that this unit is real and invariant under the homomorphism $\tau: \zeta\to\zeta^{-1}$. Then we can set $k=0$ in taking a real unit $\epsilon_i$ in $(2)$ and we have one more summand with a fixed exponent $k_j$ in the curly bracket of the sum $(5)$.
Now, the $37^{\text{th}}$ cyclotomic ring of integers is the first one with an irregular prime. The ansatz above gives the unit
$$23\equiv\zeta^{36}\cdot \varepsilon_{2}^{11}\cdot \varepsilon_{3}^{21}\cdot \varepsilon_{4}^{28}\cdot \varepsilon_{5}^{25}\cdot \varepsilon_{6}^{3}\cdot \varepsilon_{7}^{25}\cdot \varepsilon_{8}^{4}\cdot \varepsilon_{9}^{36}\cdot \varepsilon_{10}^{30}\cdot \varepsilon_{11}^{4}\cdot \varepsilon_{12}^{11}\cdot \varepsilon_{13}^{28}\cdot \varepsilon_{14}^{30}\cdot \varepsilon_{15}^{27}\cdot \varepsilon_{16}^{27}\cdot \varepsilon_{17}^{36}\cdot \varepsilon_{18}^{21}\color{blue}{\mod 37}$$
with
$$\epsilon_j:=\frac{1-\zeta^j}{1-\zeta}.$$
Numerical data suggests that if the numerators of $n$ of the Bernoulli numbers $B_2$, $B_4$, $\dots$, $B_{p-3}$ are divisible by the prime $p$ then there exist $n$ independent units that are congruent to an integer modulo $p$ and every unit that is congruent to an integer modulo p is the power of one of these $n$ units times a unit raised to the power of $p$. (When I read for the first time that Fermat's last theorem is true for the exponent $p$ if the numerators of the Bernoulli numbers $B_2$, $B_4$, $\dots$, $B_{p-3}$ are not divisible by the prime $p$ I thought fiddlesticks.) A more detailed and more rigorous proof of these calculations and the proof that this is true can be taken from the sections $16.4$ and $19.2$, here.
But this is not the only possible application of this approach. In theorem $2$ of the document Units of irregular cyclotomic fields by Washington, $1979$, he proves
There exists a basis $[\eta_2,\eta_4,\dots,\eta_{p-3}]$ for the real units modulo $[\pm 1]$ in $\mathbb Z[\zeta]$ such that
$$\eta_i\equiv a_i+b_i (1-\zeta)^{c_i}\mod (1-\zeta)^{c_i+1}$$
with $c_i=i+(p-1)u'_i$ for some integer $u'_i\ge 0$. Also
$$u'_i\le u_i=\nu_p\left(L_p(1,\omega^i)\right)$$
with the p-adic valuation $\nu_p$ normalized by $\nu_p(p)=1$ and the p-adic L-function with respect to the Teichmüller character $\omega$.
Because $(1-\zeta)^{p-1}\sim p$ we can find these units with $(5)$ simply by working modulo $\langle (1-X)^{i},m(X) \rangle$ in the cases that
- $u_i'=0$ so that $(1-\zeta)^{i}\mid p$ and
- the second factor of the class number $h_2$ is $1$ so that the fundamental system of units is just (1).
Moreover if only one unit is congruent modulo $(1-\zeta)^{p-3}$ only this unit can have an integer $u_{p-3}'>0$ because otherwise there should be a second unit that is congruent to an integer modulo $(1-\zeta)^{i+(p-1)u_i'}$, $0<i<p-3$, though there exists only one that is congruent to an integer modulo $(1-\zeta)^{p-3}$. Finally we would check whether $\eta_{p-3}\equiv a_{p-3}\pmod{(1-\zeta)^{p-1}}$ is true for a possible $u_{p-3}'>1$. By analyzing these units we obtain a good idea about the possible values of the integers $u_i'$ and we are able to compute the p-adic L-function with respect to the Teichmüller character $\omega$ by theorem $4$ of Washington's document.
Often enough the units in algebraic rings of integers are the working horses that put their shoulder to the wheel. Often enough it is at best hard to compute a fundamental system of units. As to the motivation for studying cyclotomic units: there are a lot of mathematicians out there who think that nothing is more interesting and more useless than number theory!
