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Projective Extensions of Fields
Author(s) -
Koenigsmann Jochen
Publication year - 2006
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610706022678
Subject(s) - mathematics , galois group , fundamental theorem of galois theory , pure mathematics , embedding problem , group (periodic table) , galois cohomology , galois module , conjecture , absolute galois group , field (mathematics) , discrete mathematics , chemistry , organic chemistry
Every field K admits proper projective extensions, that is, Galois extensions where the Galois group is a non‐trivial projective group, unless K is separably closed or K is a pythagorean formally real field without cyclic extensions of odd degree. As a consequence, it turns out that almost all absolute Galois groups decompose as proper semidirect products. We show that each local field has a unique maximal projective extension, and that the same holds for each global field of positive characteristic. In characteristic 0, we prove that Leopoldt's conjecture for all totally real number fields is equivalent to the statement that, for all totally real number fields, all projective extensions are cyclotomic. So the realizability of any non‐procyclic projective group as Galois group over Q produces counterexamples to the Leopoldt conjecture.