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- Tags: - #todo/untagged - Refs: - https://wstein.org/edu/2010/582e/refs/tate-galois_cohomology.pdf - Links: - Hilbert 90 - Brauer group - Tate-Shafarevich group - Unsorted/herbrand quotient

Galois cohomology

Notation

Definition

H_2

Profinite groups

Exact sequences

Applications

Proving weak Mordell-Weil

Inflation and Restriction

See also transfer, norm.

In terms of the transgression:

Cup product

Duals

See Pontrayagin dual, Cartier dual, Weil Pairing.

Local fields

For a local field: See Hasse invariant

Pairing

See also tate duality.

Relation to Artin reciprocity: [](attachments/Pasted%20image%2020220408130832.png

Examples

A use of dimension shifting

Main theorem of CFT

Proving the main theorem of class field theory: Defining the Artin map:

The local reciprocity map and the norm-residue symbol:

See quadratic reciprocity

How to compute dimensions

H_1 unramified