Field Definition (expanded) – Abstract Algebra

The field is one of the key objects you will learn about in abstract algebra. Fields generalize the real numbers and complex numbers. They are sets with two operations that come with all the features you could wish for: commutativity, inverses, identities, associativity, and more. They give you a lot of freedom to do mathematics similar to regular algebra. Today we motivate the definition of a field by looking at 6 different groups, give the formal definition, and talk about the characteristic of the field and the starting point for all fields – prime fields.

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We recommend the following textbooks:
Dummit & Foote, Abstract Algebra 3rd Edition

Milne, Algebra Course Notes (available free online)

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