Let [PSI]([W.sub.F]) denote the group of

unramified quasicharacters of [W.sub.F].

A character of [([F.sup.x]).sup.n] is called

unramified if it is trivial on [([o.sup.x]).sup.n].

Writing F for the Frobenius of Gal (Q.sup.un.sub.p])/[Q.sub.p]) one can identify Gal ([N.sub.o]/[Q.sub.p] with the subgroup of elements [F.sup.i] x [sigma] [element of] Gal ([Q.sup.un.sub.p]/Q.sub.p]) x G for which [F.sup.i] and [sigma] induce the same automorphism on the maximal

unramified extension ([N.sub.2)p] of [Q.sub.p] in [N.sub.p].

Since a [g.sup.1.sub.3] on [P.sup.1] with two points of total ramification must be

unramified everywhere else, it follows that deg([f.sub.T]) [greater than or equal to] 4.

Then [L.sub.[tau]]/k is a finite extension, [L.sub.[tau]]((t)) is the maximal

unramified subfield of [L.sub.[tau]] and [L.sub.[tau]] = [L.sub.[tau]](([tau])).

The relative genus field [k'.sup.g] of k'/k is the maximal

unramified extension of k' abelian over k.

If P corresponds to an ordinary elliptic curve then Q is

unramified over P, except that if P = 0 then the ramification degree of Q over P is 3, and if P = 1728 then the ramification degree of Q over P is 2.

The second one was obtained from this theorem studying the Swan submodule of Cl([O.sub.F][[GAMMA]]), and it implies that when p [greater than or equal to] 5, an imaginary abelian field F satisfies ([H.sub.p]) only when F/Q is

unramified at p.

Otherwise, [Sigma] is said to be nonsingular, or

unramified. Let W([Sigma]) = {w [is an element of] [W.sub.[Theta]] [where] [[Sigma].sup.w] [is approximately equal to] [Sigma]}.

We finally remark that although C(G) is rational over C for any abelian group G by Theorem 2.1, Saltman [33] gave a p-group G of order [p.sup.9] for which Noether's problem has a negative answer over C using the

unramified Brauer group [B.sub.0](G).

So [Chi] is regarded as an element of [H.sup.1] (K, Q/Z) which is

unramified on U.

For example, one may cite the study of the maximal

unramified extension of a local field or the maximal abelian extension of a global field.