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Given a number field $K$, the Dirichlet Unit Theorem tells us about the structure of the unit group $O_K^\times$. However, the proofs do not seems to give any way to explicitly write out a set of generators.

I wonder if there are special cases in which we could do that. Here are a few that comes up to me: (will edit this when references comes up in answers)

1)Quadratics fields: Solving Pell's equation.

2)Quadratic towers: we can write out the generator for a subgroup of finite index.

3)Cyclotomic fields (and its maximal real subfield): Don't know, but I am especially curious about this case.

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    $\begingroup$ Given a number field, sage (and other things) can compute the unit group for you. I think finding good algorithms is a fairly classical problem in computational number theory, but algorithms for this are well known. You might check out Section 4.9 in Cohen's "A Course in Computational Algebraic Number Theory." $\endgroup$
    – Bobby Grizzard
    May 2, 2015 at 17:12
  • $\begingroup$ also en.wikipedia.org/wiki/Cyclotomic_unit $\endgroup$
    – Bobby Grizzard
    May 2, 2015 at 17:15
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    $\begingroup$ @BobbyGrizzard: I think your comment would be a perfectly fine answer. There is an algorithm to find an explicit description of the unit group (generators etc.), hence the question should be closed. $\endgroup$
    – GH from MO
    May 2, 2015 at 17:42
  • $\begingroup$ @GHfromMO I figured the question would get shredded quickly, so I just went for a quick helpful comment. $\endgroup$
    – Bobby Grizzard
    May 2, 2015 at 20:58

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For cyclotomic fields, there are the cyclotomic units, which are formed in a very simple manner from ratios of differences of roots of unity (or for non-prime power roots of unity, just differences). An important theorem is that the set of cyclotomic units generate a subgroup $\mathcal C$ of finite index in the full group of cyclotomic units $\mathbb Z[\zeta]$. Another important result is that the index is closely related to the class number of the cyclotomic field. You'll find all of this explained in some detail in any standard book on cyclotomic fields, such as the book Introduction to Cyclotomic Fields by Larry Washington (Springer GTM) or Cyclotomic Fields I and II by Serge Lang (also a Springer GTM).

There is an analogue of cyclotomic units that are constructed quite explicitly using torsion points on elliptic curves. They are called elliptic units. See for example Modular Units by Dan Kubert and Serge Lang (in Springer's Grundlehren series).

For a modern application of cyclotomic units and elliptic units, look up the theory of Euler systems. (There are Euler systems constructed from cyclotomic units, and also Euler systems constructed from elliptic units.)

For algorithms, Cohen's book is a good place to start, but a lot has been done since it was written. There are many recent papers on computing unit groups efficiently on both classical computers and on quantum computers! One reason for this recent interest is that there are applications to certain cryptographic systems based on so-called ideal lattices.

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