A complete parametrization of cyclic field extensions of 2-power degree

Letq be a power of 2 at least equal to 8 and ζ be a primitiveq-th root of unity, and letK be any field of characteristic zero. We define the group of special projective conormsS _K as a quotient of the group of elements ofK(ζ) of norm 1:S _K is obviously trival if the groul Gal (K(ζ)/K) is cyclic. We prove that for some fieldsK, the groupS _K is finite, and it is even trivial for certain fields such as ? or ?(X ₁,...,X _m). We then prove that the groupS _K completely paramatrizes the cycle extensions ofK of degreeq. We exhibit an explicit polynomial defined over ?(T ₀,...,T _q/2) which parametrizes all cyclic extensions ofK of degreeq associated to the trivial element ofS _K. In particular, this polynomial parametrizes all cyclic extensions ofK of degreeq whenever the groupS _K is trivial.