1. 18.757 Homework 6 Solutions
1. Let Q be the additive group of rational numbers with the discrete
topology. Compute the group Xu∗ (Q) of unitary characters, the homomorphisms of Q into the group of complex numbers of absolute value 1.
(I explained in class a natural topology on Xu∗ (Q) making it a compact
abelian group.)
The description of the topology on Xu∗ (A) (in the case of any abelian group A
with the discrete topology) is this. Choose a set Γ ⊂ A of generators of A, and a
set R of defining relations: R consists of (commutative) words like
mN
γ1m1 γ2m2 · · · γN
(γj ∈ Γ, mj ∈ Z)
with the property that A is the quotient of the free abelian group generated by Γ
(that is, maps from Γ to Z with finite support) by the subgroup generated by R.
A unitary character is determined by its values on the generators, so we get an
inclusion
Y
Y
X ∗ (A) ,→
U (1)γ ,
λ 7→
λ(γ).
γ∈Γ
γ∈Γ
The image of the inclusion is clearly
)
(
Y
mN
m1
m1
mN
uγ | uγ1 · · · uγN = 1, all γ1 · · · γN ∈ R .
γ
This image is a closed subgroup in the product topology on
Y
U (1)γ ,
γ∈Γ
and that closed subspace topology is the one we choose on Xu∗ (G). The Tychonoff
theorem says that this is a compact (Hausdorff) topology.
The point of recalling all that is that the description of the topology hands
you a description of the set: choose a complex number uγ of absolute value 1 for
each generator, subject to the defining relations of A. For this problem, we need
a reasonable set of generators of Q as an additive group. The theory of partial
fractions says that every rational number has an expression
q = m0 +
N
X
mj
n
p j
j=1 j
n
(m0 ∈ Z, 0 6 mj < pj j ),
with {pj } a finite set of distinct primes, and nj a positive integer; and that this
expression is unique except for “obvious” ambiguities. (For example, if one requires
pj - mj for 1 6 j 6 N , then the expression is unique.) Restating the existence and
uniqueness of the partial fraction decomposition, we get
Proposition 1.1. The group Q is generated by γ0 = 1 and γp,n = p−n for each
prime number p and integer n > 1. We can take as relations (written additively)
γ0 − pγp,1 ,
γp,n − pγp,n+1
(n > 1)
for every prime p.
Of course we don’t need the generator γ0 , but to me it looks good to keep it.
1
2
Corollary 1.2. A unitary character of Q is given by a collection of complex numbers of absolute value 1
u0 ,
up,n
(p prime, n > 1)
subject to the relations
u0 = upp,1 ,
up,n = upp,n+1
(p prime, n > 1).
That is, we start with a single complex number u0 of absolute value 1; then, for
each prime p, we successively choose pth roots
√
√
√
up,1 = p u0 , up,2 = p up,1 , . . . up,n+1 = p up,n . . .
This corollary would have been a reasonable answer to the problem. What
follows are some more details, and some other possible answers.
Here is another way to keep track of these choices. Use the map
e : R → U (1),
e(t) = exp(2πit)
to identify U (1) with R/Z. Then the choices up,n correspond to tp,n inR/Z. The
character of Q can be specified by t0 ∈ [0, 1), and for every p a p-adic integer
a(p) = a0 (p) + pa1 (p) + p2 a2 (p) · · ·
(am (p) ∈ {0, 1, · · · , p − 1}) ;
we then define
tp,n = t0 + a0 (p) + pa1 (p) + · · · + pn−1 an−1 (p) /pn ∈ R/Z.
Corollary 1.3. A unitary character of Q is given by
t0 ∈ R/Z ' [0, 1),
and for every prime p a p-adic integer
a(p) = a0 (p) + pa1 (p) + p2 a2 (p) · · ·
(am (p) ∈ {0, 1, · · · , p − 1}) .
An advantage of this parametrization is that the nature of the choices to be
made is very concrete. A disadvantage is that the topology on Xu∗ (Q) is not so
clear. (To see why, think about the character λ corresponding to t0 = .0625 and
all am (p) = 0. Try to write some characters “close” to λ with t0 = .9375. You’ll
see that you should not choose am (2) = 0.)
Another set of generators of Q is the set of elements δN = 1/N, for N a positive
integer. The relations are
δN − M δ M N
(M, N > 0).
The conclusion is
Proposition 1.4. The group of unitary characters of Q may be identified with the
inverse limit
lim R/N Z.
←−
N >1
Accordingly there is a short exact sequence
0 → lim Z/N Z → Xu∗ (Q) → R/Z → 0;
←−
N >1
the subgroup consists of characters trivial on Z, and the quotient map is restriction
of the character to Z.
3
b and is a
The subgroup appears a lot in algebraic number theory. It’s called Z,
compact topological ring:
Y
b'
Zp ,
lim Z/N Z =def Z
←−
N >1
p
the topological product of the various compact rings of p-adic integers. Corollary
1.3 provides a set-theoretic splitting of the exact sequence of Proposition 1.4.