The interesting thing about irrational numbers is that they can't be constructed from a finite number of symbols from basic algebra. This is especially interesting when they have relationships with other irrational numbers, like the unexpected relationship between pi and e (and i) demonstrated in Euler's formula.
> 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.
No. exponentiation and square root are not algebraic operations. Only addition and multiplication are. That's kind of the whole point of this theory / subfield of mathematics.
The operations you mentioned can be found in many different subfields of mathematics, eg. real analysis, number theory, or even arithmetic (using a more broad, but a well-accepted definition). But not in algebra. It's the point of algebra to only have addition and multiplication. And it's why, for example, algebraic geometry exists (because algebraic geometers want to avoid transcendental functions like sqrt(), sin() etc.)
