Cyclotomic character

In number theory, a cyclotomic character is a character of a Galois group giving the Galois action on a group of roots of unity. As a one-dimensional representation over a ring R, its representation space is generally denoted by R(1) (that is, it is a representation χ : G → Aut_R(R(1)) ≈ GL(1, R)).

If p is a prime, and G is the absolute Galois group of the rational numbers, the p-adic cyclotomic character is a group homomorphism

where Z_p^× is the group of units of the ring of p-adic integers. This homomorphism is defined as follows. Let ζ_n be a primitive pⁿ root of unity. Every pⁿ root of unity is a power of ζ_n uniquely defined as an element of the ring of integers modulo pⁿ. Primitive roots of unity correspond to the invertible elements, i.e. to (Z/pⁿ)^×. An element g of the Galois group G sends ζ_n to another primitive pⁿ root of unity

where a_g,n ∈ (Z/pⁿ)^×. For a given g, as n varies, the a_g,n form a comptatible system in the sense that they give an element of the inverse limit of the (Z/pⁿ)^×, which is Z_p^×. Therefore, the p-adic cyclotomic character sends g to the system (a_g,n)_n, thus encoding the action of g on all p-power roots of unity.

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