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Abstract
This article examines the theory of finite extensions of the field of rational numbers, ℚ. The concepts of algebraic extensions, minimal polynomials, extension degree, and the fundamental principles of Galois theory are presented systematically. Throughout the paper, key theorems are established and their relationships with Galois groups are analyzed. The classification of finite extensions and their practical applications are also discussed. The study highlights the significance of finite field extensions in modern algebra and demonstrates their role in understanding the structure and properties of algebraic number fields.
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