Serre's conjecture II, in number theory, states that if G is a simply connected, semisimple algebraic group over a perfect field of cohomological dimension at most 2, then a certain Galois cohomology set is zero. It was proposed by Jean-Pierre Serre in 1962 as a higher-dimension equivalent of his conjecture I, and has been proven for all groups over all perfect fields, but remains open for anisotropic E6, E7 and E8 groups and the trialitarian D4 group over imperfect fields. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
StatementIf G is a simply connected, semisimple algebraic group over a perfect field F of cohomological dimension at most 2, then the Galois cohomology set H1(F, G) is zero. 1 Proposed Year Progress Toward ResolutionProven for all groups over all perfect fields; open for anisotropic E6, E7 and E8 groups and the trialitarian D4 group over imperfect fields. 1 Classification
Resolution Status Resolution Status Prize Status
Prize Status (category) Sources
1. Serre's Conjecture II (Wikipedia)
Wikimedia FoundationLead section
It was proposed by Jean-Pierre Serre in 1962
Lead section, first sentence
if G is a simply connected, semisimple algebraic group over a perfect field F of cohomological dimension at most 2, then the Galois cohomology set H1(F, G) is zero
Lead section, second paragraph
The conjecture has been proven for all groups over all perfect fields; however, it remains open for anisotropic E6, E7 and E8 groups and trialitarian D4 group over imperfect fields.
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