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My understanding of abstract algebra, and the concept of a field in particular, is that it's an abstract algebraic construction that resembles the real numbers. And that it abstracts out and captures the rules and the necessary axioms and the algebraic structure of the process of solving say an equation of the form: ax+b=0 rigorously. In a sense the definition of a field is homomorphic to the set of real numbers with the two operations.

In that light, is there an algebraic structure that abstract out the algebraic operations needed to solve an equation of the form say $2^x=3$? How to make the process of solving such an equation rigorous and how to make the necessary algebraic contex to solve such an equation? I'm assuming that the field structure is not able to solve such equation, am I right?

So my question is: is there an algebraic structure where it's possible to solve the equation $a^x+b=0$ using only that structure operations?

Edit: my suspicion that the field structure is not enough, is due to the fact that it looks like when solving the equation $a^x+b=0$, we are using more than 2 operations. We are using addition, multiplication and something new like taking nth roots or which is more problematic rising powers of transcendental numbers.

Edit2: I think my question is independent of the problem of finding a closed form solution. For example I could ask the same question about finding a solution to the equation $ax^2+bx+c=0$ using that structure operations only. The solution has a closed form, but it can't be obtained using the two operations of the field of real numbers. I used a difficult equation in the question to make the answer more general.

I0_0I
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  • Of possible interest is What Is a closed-form number? by Timothy Y. Chow (1999; JSTOR and 8 May 1998 arXiv version). – Dave L. Renfro Mar 06 '21 at 20:10
  • I think my question is independent of the problem of finding a closed form solution. For example I could ask the same question about finding a solution to the equation $ax^2+bx+c=0$ using that structure operations only. The solution has a closed form, but it can't be obtained using the two operations of the field of real numbers. I used a difficult equation in the question to make the answer more general. – I0_0I Mar 06 '21 at 20:23
  • The ruler-and-compass constructible numbers are the smallest set of numbers closed under the 4 arithmetic operations and solving of quadratic equations (with coefficients in that set) --- see the discussion and references in A detailed and elementary solution to $x^{17} = 1$. Also, Chow discusses the set $\mathbb E$ which is what you're asking about for your exponential equation. More generally, see my comment about inductive definitions here. – Dave L. Renfro Mar 06 '21 at 20:39

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