SOME THEOREMS CONCERNING PROPERLY IRREGULAR CYCLOTOMIC FIELDS

On Criteria Concerning Singular Integers in Cyclotomic Fields.

Cyclotomic Fields

Cyclotomic fields are an interesting laboratory for algebraic number theory because they are connected to fundamental problems Fermat’s Last Theorem for example and also have relatively simple algebraic properties that makes them an excellent laboratory for results in algebraic number theory. I will assume that you are familiar with basic algebraic number theory. Namely, the unique factorizatio...

Some Steinhaus type theorems over valued fields

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Some Theorems concerning Extrema of Brownian Motion With

We often use the notation ( ) to denote either ( ) or ( ). For example, ( )− ( ) denotes any one of ( )− ( ), ( )− ( ), ( )− ( ) and ( )− ( ). A point in R is called a point of local minimum (resp. local maximum) of a sample function if there exists a neighborhood of such that ( ) = ( ) (resp. ( ) = ( )). A point of either local minimum or local maximum is called an extreme-point. The following...

Classes of Cyclotomic Fields

As one will see in n◦4, these two theorems are deduced from a structure theorem for a certain group of operators. This group is constructed thus: let Xn be the p-component of the ideal class group of Kn; it is a finite abelian p-group, of order p en , which is acted opon by the Galois group G(Kn/Q) and in particular its subgroup Γn = G(Kn/K0). In passing to the projective limit over n with the ...