TITLE: Decidability of equality of expressions built using 1,+,-,*,/,^
QUESTION [27 upvotes]: Consider expressions built using number $1$, arithmetical operators $+, -, *, /$ and exponentiation ^ (in case of multiple values, the principal value is assumed, the same way as it implemented in Power function in Mathematica).
Is it a decidable problem to check if such an expression is zero?
If so, could you please point me to an algorithm that can solve this problem?
Update: I found a reference to Richardson's Theorem, that establishes undecidablity of equality in a wider set of expressions, in particular, including the logarithm and absolute value functions.

REPLY [8 votes]: The equational theory of $\langle {\bf N}, 0, 1, +, \times, \uparrow\rangle$ is decidable, but not finitely axiomatizable.