The `signature` method returns the
number of real and complex conjugate embeddings
of into
. The `unit_group`

method,
which we used above, returns the unit group
as an abstract abelian group and a homomorphism
.

Next we consider
.

Below we use the `places`

command, which returns the real embeddings
and representatives for the complex conjugate embeddings.
We use the places to define the log map , which plays such a big
role in this chapter.

Note that
, and the image lands in the
1-dimensional subspace of such that
. Also,
note that
.

Let's try a field such that . First, one with and
:

Notice that the log image of is clearly not a real multiple of the log image of (e.g., the scalar would have to be positive because of the first coefficient, but negative because of the second). This illustrates the fact that the log images of and span a two-dimensional space.

Next we compute a field with and . (A field with
is called totally real.)

A field with is called totally complex. For
example, the *cyclotomic fields*
are totally
complex, where is a primitive th root of
unity. The degree of
over
is
and , so
(assuming ).

How far can we go computing unit groups of cyclotomic fields
directly with Sage?

However, if you are willing to assume some conjectures (something
related to the Generalized Riemann Hypothesis), you can go further:

The generators of the units for
are

There are better ways to compute units in cyclotomic fields than to
just use general purpose software. For example, there are explicit
*cyclotomic units* that can be written down and generate a finite
subgroup of . See [Was97, Ch. 8], which would be
a great book to read now that you've got this far in the present book.
Also, using the theorem explained in that book, it is probably
possible to make the `unit_group`

command in Sage for cyclotomic
fields extremely fast, which would be an interesting project for a
reader who also likes to code.

William Stein 2012-09-24