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.
