Frobenius For Global Fields
In algebraic number theory, Frobenius elements are defined for extensions L/K of global fields that are finite Galois extensions for prime ideals Φ of L that are unramified in L/K. Since the extension is unramified the decomposition group of Φ is the Galois group of the extension of residue fields. The Frobenius element then can be defined for elements of the ring of integers of L as in the local case, by
where q is the order of the residue field OK mod Φ.
Lifts of the Frobenius are in correspondence with p-derivations.
Read more about this topic: Frobenius Endomorphism
Famous quotes containing the words global and/or fields:
“Much of what Mr. Wallace calls his global thinking is, no matter how you slice it, still “globaloney.” Mr. Wallace’s warp of sense and his woof of nonsense is very tricky cloth out of which to cut the pattern of a post-war world.”
—Clare Boothe Luce (1903–1987)
“The sun was like a great visiting presence that stimulated and took its due from all animal energy. When it flung wide its cloak and stepped down over the edge of the fields at evening, it left behind it a spent and exhausted world.”
—Willa Cather (1873–1947)