Field extensions

A field extension contains a smaller field and can be viewed as its vector space.

Different ways to understand it

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Bijections
A bijection is both injective and surjective and has a two-sided inverse.
Fields
A field is a commutative ring where every nonzero element has a multiplicative inverse.
Galois theory
Galois theory relates suitable field extensions to groups of automorphisms fixing the base field.
Group actions
A group action assigns compatible transformations of a set to group elements.
Rings
A ring supports addition and multiplication linked by distributive laws.