A number field is a finite degree extension of the rational numbers, $ extbf{Q}$, which means it contains elements that can be expressed as roots of polynomials with coefficients in $ extbf{Q}$. This concept is crucial as it allows the study of arithmetic properties through the lens of algebraic structures, linking to ideals, units, and class groups in a systematic way. Number fields serve as the foundation for understanding the behavior of algebraic integers, the distribution of primes, and extensions related to local and global fields.
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