Qp×

Qp× denotes the multiplicative group of nonzero p-adic numbers, where Qp is the field of p-adic numbers obtained by completing the rational numbers with respect to the p-adic absolute value for a prime p. Every nonzero x ∈ Qp can be written uniquely as x = p^n u with n ∈ Z and u ∈ Zp×, the group of p-adic units. The p-adic valuation v_p satisfies v_p(x) = n, capturing the power of p in the factorization.

As a topological group under the p-adic topology, Qp× is locally compact, totally disconnected, and abelian. It decomposes as Qp× ≅ p^Z × Zp×, separating the discrete valuation part from the unit part. The torsion subgroup of Qp× is the group of (p−1)-th roots of unity, denoted μp−1, which sits inside Zp×. For p odd, Zp× decomposes as μp−1 × (1+pZp), with (1+pZp) a pro-p group isomorphic to Zp in the topological sense. Consequently, for p odd, Qp× ≅ p^Z × μp−1 × (1+pZp). For p = 2 there is a slight variation: Z2× ≅ μ2 × (1+4Z2), with (1+4Z2) again isomorphic to Z2.

The maximal compact subgroup of Qp× is Zp×, and the subgroup p^Z captures the valuation, giving a natural split into a discrete and a compact component. This structure underpins many aspects of p-adic analysis, local field theory, and number theory, including harmonic analysis on p-adic groups and local class field theory. The notation Qp× (or Qp×) is standard for this multiplicative group.