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We study in detail the valuation theory of deeply ramified fields andintroduce and investigate several other related classes of valued fields.Further, a classification of defect extensions of prime degree of valued fieldsthat was earlier given only for the equicharacteristic case is generalized tothe case of mixed characteristic by a unified definition that workssimultaneously for both cases. It is shown that deeply ramified fields and theother valued fields we introduce only admit one of the two types of defectextensions, namely the ones that appear to be more harmless in open problemssuch as local uniformization and the model theory of valued fields in positivecharacteristic. We use our knowledge about such defect extensions to give anew, valuation theoretic proof of the fact that algebraic extensions of deeplyramified fields are again deeply ramified. We also prove finite descent, andunder certain conditions even infinite descent, for deeply ramified fields.These results are also proved for two other related classes of valued fields.The classes of valued fields under consideration can be seen as generalizationsof the class of tame valued fields. Our paper supports the hope that it will bepossible to generalize to deeply ramified fields several important results thathave been proven for tame fields and were at the core of partial solutions ofthe two open problems mentioned above.

The valuation theory of deeply ramified fields and its connection with defect extensions