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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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"Understanding" $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$
One aspect of understanding the absolute Galois group I've heard of is the 'Inverse Galois Problem'. This simply askes, Is every finite group the Galois group of some extension of Q? Its known for s …