If " K " / " k " is an extension of valued fields, then Abhyankar's inequality states that the transcendence degree of " K " / " k " is at least the transcendence degree of the residue field extension plus the " Q "-rank of the quotient of the valuation groups.
32.
In particular to establish a one-to-one correspondence between finite abelian extensions of " K " and their norm groups in an appropriate object for " K ", such as the multiplicative group in the case of local fields with finite residue field and the idele class group in the case of global fields, as well as to describe those norm groups directly, e . g ., such as open subgroups of finite index.
33.
For example, the field ! ( i ) of Gaussian rationals, if equipped with the valuation w given by w ( 2 + i ) = 1 ( and w ( 2-i ) = 0 ) is formally 5-adic ( the place " v " = 5 of the rationals splits in two places of the Gaussian rationals since X ^ 2 + 1 factors over the residue field with 5 elements, and " w " is one of these places ).
34.
In fact, they agree in the sense that when " A " is the ring of polynomials over some algebraically closed field " k ", the maximal ideals of " A " are ( as discussed in the previous paragraph ) identified with " n "-tuples of elements of " k ", their residue fields are just " k ", and the " evaluation " maps are actually evaluation of polynomials at the corresponding " n "-tuples.
