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Let K be a number field and n=[K: Q]=r₁+2r₂ where r₁ is the number of real embeddings K \hookrightarrow C , and r₂ the number of pairs of complex embeddings K \hookrightarrow C . Then

OK* ≅ μ(K) ×Zr1+r2−1,

where μ(K) is the set of roots of unity in K .
Put another way, this means that there are r=r₁+r₂−1 "fundamental units" ε₁, …, ε_r such that any unit u ∈ O_K^* can be written uniquely as

u = ζε1n1 …εrnr

for some ζ ∈ μ(K) .