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  "body": "\n## Summary\n\nThe cohomological dimension of a field $K$ is defined through its [[Galois cohomology|9MPCZ3EAJM]], see the [[definition|XKMBKAN1MR]].\nIt is the largest degree in which this cohomology with torsion coefficients can be nonzero.\nIt is the [[cohomological dimension|EF4ZWACHD8]] of the [[absolute Galois group|PGKWBJDTKX]] of $K$, and it can be computed one prime at a time.\nSeparably closed fields have dimension $0$, finite fields have dimension $1$, and $p$-adic fields have dimension $2$.\nIt controls which Galois cohomology groups can appear in arithmetic questions over the field.\n\n## Explanations\n\n### L1 human\n\nGalois cohomology attaches to a field a sequence of groups $H^0, H^1, H^2, \\ldots$.\nFor many fields these groups vanish above some degree, and that degree is the cohomological dimension.\nA field in which every polynomial splits has nothing above degree $0$.\nA finite field has nothing above degree $1$, and the field of $p$-adic numbers has nothing above degree $2$.\nFor the real numbers the groups never stop, so the dimension is infinite.\n",
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