A class formation is a formation such that for every normal layer E/F
- H1(E/F) is trivial, and
- H2(E/F) is cyclic of order |E/F|.
In practice, these cyclic groups come provided with canonical generators uE/F ∈ H2(E/F), called fundamental classes, that are compatible with each other in the sense that the restriction (of cohomology classes) of a fundamental class is another fundamental class. Often the fundamental classes are considered to be part of the structure of a class formation.
A formation that satisfies just the condition H1(E/F)=1 is sometimes called a field formation. For example, if G is any finite group acting on a field A, then this is a field formation by Hilbert's theorem 90.
Read more about Class Formation: Examples of Class Formations, The First Inequality, The Second Inequality, The Brauer Group, Tate's Theorem and The Artin Map, The Takagi Existence Theorem, Weil Group
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