Galois theory

Galois theory relates suitable field extensions to groups of automorphisms fixing the base field.

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Field extensions
A field extension contains a smaller field and can be viewed as its vector space.
Group actions
A group action assigns compatible transformations of a set to group elements.
Binary operations
A binary operation combines two elements into one in a specified set.
Cosets and Lagrange's theorem
Cosets partition a finite group, so a subgroup's order divides the group's order.
Cyclic groups
A cyclic group is generated by repeated powers of a single element.