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Galois theory, named after Évariste Galois, provides a connection between field theory and group theory. Using Galois theory, certain problems in field theory can be reduced to group theory, which is, in some sense, simpler and better understood.
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Galois theory and algorithms
As Noam Elkies wrote, this statement was made in the context of finding eigenvalues. The full paragraph reads "We not only show the existence of the Jordan canonical form, but also derive an algorithm …