Cyclotomic units are a special class of elements in the field of algebraic number theory, particularly within the context of cyclotomic fields. Cyclotomic fields are extensions of the rational numbers obtained by adjoining a primitive \( n \)-th root of unity, denoted as \( \zeta_n \), to the rationals \( \mathbb{Q} \).
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Cyclotomic units include roots of unity and suitable ratios in cyclotomic fields. Compatible roots make these ratios norm-compatible along cyclotomic towers. Their closed norm-limit module is a concrete submodule of global units, and its local images generate the power series in the Iwasawa main conjecture.