Showing papers on "Generic polynomial published in 1993"
TL;DR: In this paper, the existence of the Galois extensionM/L/k such that the canonical projection Gal(M/k)→Gal(L/K) coincides with the given homomorphismj:E→G and that M/L is unramified.
Abstract: LetL/k be a finite Galois extension with Galois groupG, and\(1 \to A \to E\xrightarrow{j}G \to 1\) a group extension. We study the existence of the Galois extensionM/L/k such that the canonical projection Gal(M/k)→Gal(L/k) coincides with the given homomorphismj:E→G and thatM/L is unramified.
5 citations
TL;DR: In this paper, the authors define the group of special projective conormsS fixme K as a quotient of a group of elements of K(ζ) of norm 1:S isEnabled K is a trival if the groul Gal (K(ϵ)/K) is cyclic.
Abstract: 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 ℚ or ℚ(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 ℚ(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.
5 citations