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I am researching about the Galois cohomology of unit group of a Galois sextic number field $L$ with Galois group $S_3$, the symmetric group on $3$ symbols.

I want to find that "Is there a explicit way for direct computation of cocycles and coboundaries in the first group cohomology of $S_3$ with coefficients in $U_L$, $H^1(S_3, U_L)$?"

I read some books about Galois cohomology, i.e. J. Serre or Neukirch or Berhuy and etc. But unfortunately I don't find a good example in these books, at least I wish to find that "What is the order of Galois cohomology group of $H^1(S_3, U_L)$?"

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    $\begingroup$ Possible duplicate of What is known about first cohomology of the units in a number field? $\endgroup$
    – SashaP
    Sep 17, 2016 at 14:25
  • $\begingroup$ It sounds like you're more interested in working out a specific explicit example than the kind of general statements in the linked question (so I think this question is not a duplicate). Have you tried computing a $\mathbf{Z}$-generating set for your unit group (which isn't that large; it will be isomorphic to $\mathbf{Z}^2 \times $(finite) by Dirichlet's unit theorem) and then writing down how $S_3$ acts in that basis? Then writing down cocycles and coboundaries is "just" linear algebra. $\endgroup$
    – David Loeffler
    Sep 19, 2016 at 8:34

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