Spectral norms on valued fields

Let (K, \.\) be a perfect valued field, (K) over bar be an algebraic closur e of K, \.\ be an extension of \.\ to (K) over bar, G = Gal((K) over bar /K ) and parallel tox parallel to = sup {\ sigma (x)\ \ sigma epsilonE G} be t he G-spectral norm on (K) over bar. Let K subset of L subset of (K) over ba r be an algebraic extension of K and (L) over tilde be the completion of L relative to parallel to.parallel to. We associate to any element x epsilon (L) over tilde a real number omega (x) and prove that if omega (x) > 0, for all x in (L) over tilde\(K) over tilde, then (L) over tilde = boolean OR { (E) over tilde \ K subset of E subset of L, [E : K] < infinity} and (L) ove r tilde is a zero-dimensional regular ring. We show that (L) over tilde boo lean AND (K) over bar = L and prove that (L) over tilde is algebraic over ( K) over tilde (with some additional conditions on K and L). We give a Galoi s type correspondence between the set of all closed K-subalgebras of (L) ov er tilde and the subfields of L. We prove that (Q)congruent to is an algebr aic closed and zero-dimensional regular ring.