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Is there any theory (analogous to Galois theory) for solving equations with irrational exponents like:

$ x^{\sqrt{2}}+x^{\sqrt{3}}=1$

?

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    @J.Mangaldan: From the wiki article I linked, it says "The (Newton) method will usually converge, provided this initial guess is close enough to the unknown zero". This implies to me that this method will (under the right conditions) generate a sequence of real numbers which converges to the desired solution. I would call this an "analytic solution" since I think of "analysis" as being primarily about converging sequences. Perhaps there is something I am misunderstanding here. I call the finite iterations "approximations", not the limit (if it exists). Am I wrong to use this terminology?2010-08-30

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The study of such equations is not "abstract algebra" as it is usually understood. The reason is that to even define the function $x^{\sqrt{2}}$, for example, requires analysis; one has to prove certain properties of $\mathbb{R}$ to ensure that such a function exists. This is in marked contrast to the case of integer or rational powers, where one has a purely algebraic definition and the background theory is equational. To define the function $x^{\sqrt{2}}$ one has to either define $e^x$ and the logarithm or consider a limit of functions $x^{p_n}$ where $p_n$ form a sequence of rational approximations to $\sqrt{2}$, and this is irreducibly non-algebraic stuff.

In particular, while polynomials can be studied in an absurdly general setting, transcendental equations like those you describe are more or less restricted to $\mathbb{R}$ (or $\mathbb{C}$ if you really want to pick a branch of the logarithm). The LHS is an increasing function of $x$, so there is at most one root, which probably one can really only compute numerically if it exists. (Its nonexistence can be ruled out by computing local minima in $(0, 1)$.)

This is another question which touches on a theme which has come up several times on math.SE, which is that exponentiation should really not be thought of as one operation. Instead, it is a collection of related operations with various degrees of generality and applicability which happen to share the same algebraic properties, and one should not infer too much about how similar these operations are.

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    Fair point; I'll modify the wording.2010-08-30