Lindemann-Weierstrass theorem

The Lindemann-Weierstrass theorem states that if you take any non-zero algebraic number (a number that solves a polynomial with rational coefficients) and exponentiate it (calculate e to that power), the result will always be a transcendental number—meaning it cannot be the root of any polynomial equation with rational coefficients. This theorem shows that some fundamental mathematical constants, like *e* and certain expressions involving exponentials of algebraic numbers, are inherently beyond algebraic solutions, highlighting their deep and non-repeating complexity within mathematics.