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Q×

Q× denotes the multiplicative group of nonzero rational numbers. It is defined as Q× = { q ∈ Q : q ≠ 0 } with the operation of multiplication. The group is abelian, since the multiplication of rational numbers is commutative.

Every element q ∈ Q× can be uniquely written as q = ε ∏_{p prime} p^{n_p}, where ε ∈ {−1, 1}

Properties include that Q× is infinite and countable, and it is not finitely generated. It is a

Examples: 6 lies in Q× and factors as 2^1 · 3^1; 1/8 equals 2^(−3); −14 equals (−1) · 2

Context: As the unit group of the field Q, Q× is a standard example in algebra of

is
the
sign
and
n_p
∈
Z
with
only
finitely
many
nonzero.
This
yields
a
canonical
decomposition
Q×
≅
(⊕_{p
prime}
Z)
⊕
Z/2Z:
a
free
abelian
group
of
countable
rank
together
with
a
two-element
torsion
subgroup
{±1}.
Equivalently,
the
positive
part
Q×+
is
a
free
abelian
group
on
the
primes,
generated
by
the
primes
with
integer
exponents.
subgroup
of
the
real
numbers
under
multiplication,
and
as
a
topological
group
it
inherits
the
subspace
topology
from
R×.
·
7.
a
free
abelian
group
of
countable
rank
up
to
a
finite
torsion
component.
It
arises
in
discussions
of
arithmetic,
divisors,
and
valuations
in
the
rational
numbers.