D. J. Bernstein
More mathematics
Math 514, Number Theory I, Fall 1995
Thoughts on teaching algebraic number theory
Dirichlet's unit theorem
One can prove a computationally effective form of
Dirichlet's unit theorem with no extra effort.
Here's how.
(Notation: t is the desired unit rank.)
- Prove that unit log vectors are discrete.
- Construct units u1, u2, ..., ut,
with log vectors that are independent over the reals.
- Observe that
the unit group is generated by u1, u2, ..., ut together with
all units w such that,
for each embedding sigma,
|log|sigma(w)|| is at most half of
|log|sigma(u1)|| + |log|sigma(u2)|| + ... + |log|sigma(ut)||.
By discreteness, there are only finitely many such w.
Hence the unit group is finitely generated.
By discreteness again, the unit group has rank t.
Here is what you see in the usual textbooks:
- Prove that unit log vectors are discrete.
- Prove that any discrete subgroup of Euclidean space is a lattice
of some rank.
First select a maximal R-independent subset, say x1, x2, ..., xr.
Then observe that the subgroup is generated by x1, x2, ..., xr
together with all elements y such that,
for each coordinate i,
|y_i| is at most half of
|x1_i| + |x2_i| + ... + |xr_i|.
By discreteness, there are only finitely many such y.
Hence the subgroup is finitely generated.
By discreteness again, it has rank exactly r.
- Construct the right number of units, u1, u2, ..., ut,
with log vectors that are independent over the reals.
Conclude that the unit group has rank t.
My approach skips the step of selecting a maximal R-independent set
of some unknown size---I've already
constructed the right number of independent units!
My approach also makes clear that you can actually compute the unit group
(and thus the class group) of any number field;
this fact is notably absent from the textbooks.
Rings
Two conventions to establish
in any introductory course on algebraic number theory,
algebraic geometry, or commutative algebra:
``In modern mathematics, all rings contain 1.
In this course, all rings are commutative.''
(I never needed to explicitly refer to a noncommutative ring.
If I had, I would have called it a ``thing.'')
With those conventions:
A ring is any object with 0 1 + - * that satisfies
every identity satisfied by the integers.
This fact should be stated in algebra courses,
and some students figure it out on their own,
but it can't hurt to say it again.
Integrally closed
Don't say ``integrally closed.'' Say ``nonsingular.''
I used ``integrally closed'' in this course;
I didn't realize how confusing it would be for the students
that A can be integral over R and bigger than R
even if R is integrally closed.