Proposition 11.8. Assuming that:

  • OK a Dedekind domain

  • L∕K a finite Galois extension

  • 0≠P⊆OL a prime ideal

  • P|𝔭⊆OK

Then
  • (i) LP∕K𝔭 is Galois.
  • (ii) There is a natural map
    res⁡:Gal⁡(LP∕K𝔭)→Gal⁡(L∕K)

    which is injective and has image GP.