p-adic fields are extensions of the field of p-adic numbers, which are constructed using a unique valuation that measures the size of numbers based on their divisibility by a prime number p. These fields play a significant role in number theory and algebraic geometry, particularly in understanding local properties of schemes and the behavior of algebraic varieties over finite fields. The study of p-adic fields intertwines with various concepts, such as local fields, ramification theory, p-adic Hodge theory, and Tate modules.
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