Let K be a

finite extension of either Qp or Fp((t)).

To keep the world as we see it now (inertia, mass, etc.); all Mach based cosmologies must have the universe started at a finite past and also must have a

finite extension. So this way the contradiction of infinity is not solved.

It is well known that the field of formal power series over finite fields has a lot of properties in common to number fields (the

finite extension of Q).

Leibniz describes here two "counterfactuals": we have rules which allow us to treat matter as if it were extended yet indivisible and rules which allow us to treat

finite extension as if it were composed of infinitely small extended metaphysical beings.

Let L = K(Q), then L/K is a

finite extension and L [subset] [K.sub.v] so, in particular, v is unramified in L and, a fortiori, L/K is separable.

Given any

finite extension F of the rational field Q in the complex field C, we denote by Nf the norm map from F to Q.

A homogeneous polynomial [Mathematical Expressions Omitted] is called decomposable if it can be factorized into linear facotrs over some

finite extension field G of K.

(iii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are cohomologically coprime for some prime number ' and for every

finite extension F' of F.

We study the subfields of division algebras via index reduction, using the fact that a

finite extension L/k of degree m is a subfield of a k-division algebra of index n if and only if m [vertical bar] n and ind ([D.sup.L]) = n/m.

1) If the (classical) cohomological Hasse principle holds for G over every

finite extension L [subset] [L.sub.n] [subset] k, then the same holds for G over k.

Let K be a global field (i.e.,

finite extension of Q or algebraic function field in one variable over a finite field) and [G.sub.K] its absolute Galois group.

[subset] [L.sub.i] [subset] [L.sub.i + 1] [subset] k is a tower of increasing

finite extensions of L.