More precisely it depends on the choice of a separable algebraic closure of the residue field of " A ", and automorphisms of this separable algebraic closure correspond to automorphisms of the corresponding strict Henselization.
12.
A Cohen ring is a field or a complete characteristic zero discrete valuation ring whose maximal ideal is generated by a prime number " p " ( equal to the characteristic of the residue field ).
13.
The above implies that there is an equivalence of categories between the finite unramified extensions of a local field " K " and finite separable extensions of the residue field of " K ".
14.
From the geometric point of view, " n "-dimensional local fields with last finite residue field are naturally associated to a complete flag of subschemes of an " n "-dimensional arithmetic scheme.
15.
If is a prime of lying over, that is unramified means by definition that the integers of modulo, the residue field of, will be a finite field of order extending the residue field of where is the degree of.
16.
If is a prime of lying over, that is unramified means by definition that the integers of modulo, the residue field of, will be a finite field of order extending the residue field of where is the degree of.
17.
Additionally, the conductor can be defined when " L " and " K " are allowed to be slightly more general than local, namely if they are complete valued fields with quasi-finite residue field.
18.
Sometimes the valuation ring " K " itself is excluded, and sometimes the points are restricted to the zero-dimensional valuation rings ( those whose residue field has transcendence degree zero over " k " ).
19.
In mathematics, "'perfectoid "'objects occur in the study of problems of " mixed characteristic ", such as local fields of characteristic zero which have residue fields of characteristic prime " p ".
20.
For " p "-adic fields the Weil group is a dense subgroup of the absolute Galois group, and consists of all elements whose image in the Galois group of the residue field is an integral power of the Frobenius automorphism.
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