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Algebraic Number Theory - Definitions

Briefly define a group.
Briefly define a ring.
Briefly define a field.

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A non-empty set S is said to form a group under a binary operation (*) if it satisfies the following properties-

(1) S is closed with respect to (*) [Closure property]

(2) (*) is associative on S [Associative property]

(3) S has an identity element under (*) [Identity ...

Solution Summary

Neat definitions with examples are provided.