Algebraic Number Theory
An adèle is a mathematical object that arises in the study of algebraic number theory, representing an element of the adèle ring, which is constructed from the completion of a global field at all places. Adèles provide a powerful way to combine local information from various completions of number fields, allowing for the understanding of arithmetic properties in a unified manner. They are particularly useful in class field theory, where they help facilitate connections between different number fields and their Galois groups.
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