Transcendental Number

In mathematics, a transcendental number is a real or complex number that is not algebraic—that is, it is not a root of a non-zero polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e. Though only a few classes of transcendental numbers are known (in part because it can be extremely difficult to show that a given number is transcendental), transcendental numbers are not rare. Indeed, almost all real and complex numbers are transcendental, since the algebraic numbers are countable while the sets of real and complex numbers are both uncountable. All real transcendental numbers are irrational, since all rational numbers are algebraic. The converse is not true: not all irrational numbers are transcendental; e.g., the square root of 2 is irrational but not a transcendental number, since it is a solution of the polynomial equation x2 − 2 = 0.
== Tossup Questions ==
# The set of numbers with this property was divided into S, T, and U varieties by Kurt Mahler. If A does not have this property and B is irrational, Hilbert's seventh problem asks whether A raised to the B always has this property. A man who proved that this property is possessed by "e to the a," when a has this property, is the first namesake of the Lindemann-Weierstrass theorem, which used Euler's identity to show that e and another number possess it. For 10 points, the non-algebraics, such as pi, have what property, meaning they are not the solution of any polynomial?