How to generalize a unit formula like that of quadratic integer rings to non-abelian extensions? Let $k$ be the quadratic integer ring $\mathbb{Z}(\sqrt{p})$ or $\mathbb{Z}(\frac{-1+\sqrt{p}}{2})$, whenever appropriate, and where $p$ is a positive squarefree integer. There exists a power of the fundamental unit $\epsilon_{k}$ that is expressible in terms of a product of cyclotomic integers;
$\epsilon^{h}_{k}=\displaystyle\prod_{(l, \Delta)=1}\big(1-\zeta^{l}_{\Delta}\big)^{\chi_{\Delta}(l)}$
where $h$ is some integer, $\Delta$ is the absolute value of the discriminant of $k$, $\zeta^{l}_{\Delta}$ is a $\Delta$'th root of unity, and $\chi_{\Delta}$ is the appropriate quadratic character of modulus $\Delta$. And the product runs over $\Delta \geq l \geq 1$
In general, the units of the integer ring of an abelian extension of $\mathbb{Q}$ can be given as a particular quotient of these integers sitting in some cyclotomic field (I think?).
The question is, does there exist a similar algebraic solution for all of the generators of the units of $\mathcal{O}_{K}$, where $K$ is a Galois extension of $\mathbb{Q}$ but not abelian? Even for $K$ that isn't Galois?
 A: This formula for real quadratic fields is a special case of the analytic
class number formula. This states that the value of the residue of the Dedekind zeta function $\zeta_K(s)$ of the number field $K$ at $s=1$
is a fairly predictable number (involving the discriminant of $K$)
times $h_KR_K$. Here $h_K$ is the class number of $K$ and $R_K$ is the regulator.
The regulator of $K$ is the absolute value of a $t\times t$ matrix
made up of logarithms of the absolute values of a system of fundamental
units. Here $t$ is the rank of the group of units. When $K/\Bbb Q$
is Abelian, the residue factors as a product of various $L(\chi,1)$
for Dirichlet characters $\chi$. There are fairly explicit formulae
for these. When $t=1$, as in the real quadratic case, the regulator
is essentially the logarithm of the fundamental unit. Therefore
we get a formula for the fundamental unit in this case.
If $K/\Bbb Q$ is not Abelian, we don't get a nice formula for the
residue of $\zeta_K$. If $t>1$ the regulator, by itself, doesn't
give you the logs of the fundamental unit. Things just got more
complicated....
A: There are in fact similar formulas for special classes of fields coming from the theory of complex multiplication. The first example goes back to Dedekind, who studied class numbers of pure cubic number fields. His ideas were worked out using class field theory by Hasse's student Curt Meyer (whose 
book is close to being illegible for all but a few who are perfectly at home in CM); Ken Nakamula used such class number formulas for actually computing class numbers. The keyword to look for is "elliptic units".
In some sense, the Stark conjectures also supply you with formulas that may be seen as generalizations of the cyclotomic formula you mentioned.
