A valuation on a field K extends to an algebraic extension L in finitely many ways, and each extension w/v carries numerical invariants: the ramification index e, the residue degree f, the local degree n, and the defect d = n/(ef), which together satisfy the fundamental equality [L:K] = sum over w|v of d*e*f. Working through group actions rather than ideal theory, the automorphism group of a normal extension acts transitively on the set of extending valuations, and the stabilizer chain (decomposition, inertia and ramification groups) cuts out a tower whose successive degrees are exactly the number of extensions, the separable residue degree, the tame ramification index and the wildness index. Students learn how to call an arbitrary algebraic extension immediate, unramified, tame, local, totally ramified or totally wild by which invariants equal one, to test each property against automorphism groups and fundamental sets rather than only in the Galois case, and to locate the unique maximal immediate/unramified/tame and unique minimal local/totally-ramified/totally-wild subextensions.
A valuation ring of a field K is a subring O such that every nonzero x in K has x or x-inverse in O; it is local, its residue field is O/m, and K*/O* is an ordered abelian group called the value grou…