Suppose you have an unlimited quantity of the number one, and the operators plus, minus, multiply, divide, and power. Consider the (countable) set $S$ you generate by combining these:
Using just one and plus, you can construct the natural numbers.
Using minus, you can construct the integers.
Using divide, you can construct the rational numbers.
Using power, you can construct nth roots like $2^{1/2} = \sqrt{2}$
So far, so good. However, you can now go further and construct things like $5^\sqrt{2}$ and far more bizarre things. Questions:
Did Galois show $S$ is a proper subset of the algebraic numbers? I know he showed arbitrary 5th-degree polynomials don't have "closed- form" solutions, but I believe his definition of "closed form" was more limited.
If not, let $T$ be the set of numbers Galois considers "closed form". Are there members of $S-T$ that solve high order polynomials?
$S$ seems like an "obvious" set to me. Does it have a name, and do people study it?
I chose $5^\sqrt{2}$ as a "random example": it seems obvious that it's non-algebraic, but I can't seem to prove it.