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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} 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.

Properties include that Q× is infinite and countable, and it is not finitely generated. It is a subgroup of the real numbers under multiplication, and as a topological group it inherits the subspace topology from R×.

Examples: 6 lies in Q× and factors as 2^1 · 3^1; 1/8 equals 2^(−3); −14 equals (−1) · 2 · 7.

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