Rings, groups, and fields all feel similar. What are the differences between them, both in definition and in how they are used?
They should feel similar! In fact, every ring is a group, and every field is a ring. A ring is an abelian group with an additional operation, where the second operation is associative and the distributive property make the two operations "compatible".
A field is a ring such that the second operation also satisfies all the properties of an abelian group (after throwing out the additive identity), i.e. it has multiplicative inverses, multiplicative identity, and is commutative.
