section_path: Irrationality and transcendence
section_name: Irrationality and transcendence
paragraph_text: Irrationality and transcendence
  
  If b is a positive real algebraic number, and x is a rational number, then b is an algebraic number. This results from the theory of algebraic extensions. This remains true if b is any algebraic number, in which case, all values of b (as a multivalued function) are algebraic. If x is irrational (that is, not rational), and both b and x are algebraic, Gelfond–Schneider theorem asserts that all values of b are transcendental (that is, not algebraic), except if b equals 0 or 1.
  
  In other words, if x is irrational and b ∉ { 0, 1 }, {\displaystyle b\not \in \{0,1\},} then at least one of b, x and b is transcendental.