> an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and satisfying the axioms implied by "vector space" and "bilinear"
Historically though, the word "algebra" was used more broadly, and the further into the past you go, the more vague this term becomes. But, today, if you ask a mathematician, the definition above is how they would immediately understand algebra, and other kinds of algebras would need a qualification, eg. "linear algebra" or "abstract algebra" etc.
Another way to look at this is to say that various subfields of mathematics that are called "algebra" are studies of particular kinds of algebra (from the first definition). And so they will still have all the same elements: a set (with some restrictions on it), a multiplication and addition.
It could be surprising that so few basic elements give rise to such a rich field, but that's how math is... In a way, the elements you work with act more as constraints rather than extra dimensions. So, theories with very few basic elements tend to capture more stuff and be richer in terms of theorems than theories with more basic elements.
You’re confusing “algebra” with “an algebra.” You’re misunderstanding the terms here. For a simple example, group theory is absolutely a branch of algebra, and a group is not “an algebra”
I don't see how any of that matters to answering the question. Anything that's labeled "algebra" will have addition, multiplication and a field over which those operations are defined. This is the whole point of the term.
Transcendental functions s.a. sqrt() are to algebra like the trolley problem is to physics: deliberately excluded from the domain of discourse.
> an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and satisfying the axioms implied by "vector space" and "bilinear"
from: https://en.wikipedia.org/wiki/Algebra_over_a_field
Historically though, the word "algebra" was used more broadly, and the further into the past you go, the more vague this term becomes. But, today, if you ask a mathematician, the definition above is how they would immediately understand algebra, and other kinds of algebras would need a qualification, eg. "linear algebra" or "abstract algebra" etc.
Another way to look at this is to say that various subfields of mathematics that are called "algebra" are studies of particular kinds of algebra (from the first definition). And so they will still have all the same elements: a set (with some restrictions on it), a multiplication and addition.
It could be surprising that so few basic elements give rise to such a rich field, but that's how math is... In a way, the elements you work with act more as constraints rather than extra dimensions. So, theories with very few basic elements tend to capture more stuff and be richer in terms of theorems than theories with more basic elements.