Minimal pairs, inertia degrees, ramification degrees and implicit constant fields

An extension (K(X)|K, v) of valued fields is said to be valuation transcendental if we have equality in the Abhyankar inequality. Minimal pairs definition are fundamental objects investigation extensions. In this article, associate a uniquely determined positive integer with extension. This defined via chosen minimal pair definition, but it later shown independent choice. Further, show that encodes important information regarding implicit constant field v).