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Thus when defining formal Laurent series, one requires Laurent series with only finitely many negative terms. Similarly, the sum of two convergent Laurent series need not converge, though it is always defined formally, but the sum of two bounded below Laurent series (or any Laurent series on a punctured disk) has a non-empty annulus of convergence.
The formal Laurent series form the ring of formal Laurent series over , denoted by (()). [ b ] It is equal to the localization of the ring R [ [ X ] ] {\displaystyle R[[X]]} of formal power series with respect to the set of positive powers of X {\displaystyle X} .
The formal Laurent series over a finite field: the ring of integers of F q ((T)) is the ring of formal power series F q [[T]]. Its maximal ideal is (T) (i.e. the set of power series whose constant terms are zero) and its residue field is F q. Its normalized valuation is related to the (lower) degree of a formal Laurent series as follows:
In mathematics, local class field theory, introduced by Helmut Hasse, [1] is the study of abelian extensions of local fields; here, "local field" means a field which is complete with respect to an absolute value or a discrete valuation with a finite residue field: hence every local field is isomorphic (as a topological field) to the real numbers R, the complex numbers C, a finite extension of ...
Local class field theory of characteristic p>0: The module A is the group of units of the separable algebraic closure of the field of formal Laurent series over a finite field, and G is the Galois group.
This completion can be described as the field of formal Laurent series over . It is a non-Archimedean ordered field. Sometimes the term "complete" is used to mean that the least upper bound property holds, i.e. for Dedekind-completeness. There are no Dedekind-complete non-Archimedean ordered fields.
For each formal Frobenius series solution of =, must be a root of the indicial polynomial at , i. e., needs to solve the indicial equation =. [ 1 ] If ξ {\displaystyle \xi } is an ordinary point, the resulting indicial equation is given by α n _ = 0 {\displaystyle \alpha ^{\underline {n}}=0} .
If is a non-trivial automorphism of complex numbers (such as the conjugation), then the resulting ring of Laurent series is a noncommutative division ring known as a skew Laurent series ring; [11] if σ = id then it features the standard multiplication of formal series.