Cohn's irreducibility criterion

In mathematics, a cyclotomic unit is a unit of an algebraic number field which is the product of numbers of the form (ζTemplate:Su − 1) for ζTemplate:Su an n^th root of unity and 0 < a < n. Note that if n is the power of a prime ζTemplate:Su − 1 itself is not a unit; however the numbers (ζTemplate:Su − 1)/(ζTemplate:Su − 1) for (a, n) = 1, and ±ζTemplate:Su generate the group of cyclotomic units in this case (n power of a prime).

The cyclotomic units form a subgroup of finite index in the group of units of a cyclotomic field. The index of ths subgroup of real cyclotomic units (those cyclotomic units in the maximal real subfield) within the full unit group is equal to the class number of the maximal real subfield of the cyclotomic field.

Note also that if n is a composite number, the subgroup of cyclotomic units generated by (ζTemplate:Su − 1)/(ζTemplate:Su − 1)with (a, n) = 1 is not of finite index in general.

The cyclotomic units satisfy distribution relations. let a be a rational number prime to p and let g_a denote exp(2πia)−1. Then for a≠ 0 we have $\prod _{pb=a}g_{b}=g_{a}$.

Using these distribution relations and the symmetry relation ζTemplate:Su − 1 = -ζTemplate:Su (ζTemplate:Su − 1) a basis B_n of the cyclotomic units can be constructed with the property that B_d ⊆ B_n for d | n.

See also

• Elliptic unit
• Modular unit

References

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