The fundamental theorem of Galois theory establishes a deep connection between field extensions and group theory, specifically relating the structure of a field extension to the properties of its Galois group. It states that there is a correspondence between the subfields of a field extension and the subgroups of its Galois group, allowing for an understanding of how the roots of polynomials behave under automorphisms. This connection is crucial when applying Galois theory to number fields and understanding how different extensions relate to each other.
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