Let K be a nonarchimedean local field. As explained in the previous lecture, to complete our proof the existence theorem of local class field theory (Theorem 27.8), which states that every finite index open subgroup of K× is a norm group, we need to construct a system of totally ramified abelian extensions Kπ,n/K and a homomorphism θπ : K × → Gal(Kab π /K) satisfying the hypotheses of Theorem 3...

In this paper‎, ‎the existence and uniqueness of the ‎solution of a nonlinear fully fuzzy implicit integro-differential equation‎ ‎arising in the field of fluid mechanics is investigated. ‎First,‎ an equivalency lemma ‎is ‎presented ‎by‎ which the problem understudy ‎is ‎converted‎ to ‎the‎ two different forms of integral equation depending on the kind of differentiability of the solution. Then...

Summary This is the first part of a two-part article formalizing existence and uniqueness algebraic closures using Mizar system [1], [2]. Our proof follows Artin’s classical one as presented by Lang in [3]. In this we prove that for given field F there exists extension E such every non-constant polynomial p ∈ [ X ] has root . applies Kronecker’s construction to each \F simultaneously. To do so ...

We show that a Z-cyclic triplewhist tournament TWh(v) exists whenever v =p] . . . . p~ where the primes p~ are -5(mod8) , p~>29. The method of construction uses the existence of a primitive root ~o of each such Pi (~61) such that ~o2+eo+ 1 are both squares (modpi).

A field K is called an n-dimensional local field if there is a sequence of fields kn, . . . , k0 satisfying the following conditions: k0 is a finite field, ki is a complete discrete valuation field with residue field ki−1 for i = 1, . . . , n, and kn = K . In [9] we defined a canonical homomorphism from the n th Milnor group Kn(K) (cf. [14]) of an n-dimensional local field K to the Galois group...