One-sided reciprocity for cyclotomic pseudo-units #
This file isolates the arithmetic core of one-sided Kummer reciprocity in the canonical Kummer presentation of an inverse-cyclotomic extension. The symbol used below is the raw finite-prime Frobenius product, followed by the canonical Kummer coordinate; no class-field-theory or reciprocity hypothesis is bundled into its definition.
The raw Kummer/Frobenius product is the cyclic Artin product followed by the coordinate supplied by the canonical Kummer presentation.
Vanishing of the raw Kummer/Frobenius product is equivalent to vanishing of the same finite-prime product in the cyclic coordinate.
The divisor root already defined by the canonical Kummer presentation
has the expected p-th power whenever all radicand exponents are divisible
by p. This formulation uses exactly the pseudo-unit hypothesis, without
an unramifiedness assumption.
Unit-valued form of the pseudo-unit divisor-root identity.
Every value of the raw canonical Kummer/Frobenius symbol has exponent
dividing p, because its target is the group of p-th roots of unity.
Consequently, the raw canonical Kummer/Frobenius symbol kills every
p-th power in the fractional-ideal group.
The pseudo-unit divisor condition alone makes the raw symbol vanish on the principal fractional ideal of the canonical radicand. The missing one-sided reciprocity theorem is the distinct assertion with an arbitrary principal ideal in the denominator slot.