Lindemann–Weierstrass theorem

The Lindemann–Weierstrass theorem states that exponentials of distinct algebraic numbers are linearly independent over the algebraic numbers. It implies that the number π is transcendental.

Algebraic number: A complex number that is a root of a nonzero polynomial with rational coefficients. The theorem's inputs are distinct algebraic numbers, and its coefficients come from this same field.
Rational number: A number expressible as a ratio of two integers with a nonzero denominator. Rational numbers form the coefficient field used in defining algebraic numbers.
Charles Hermite: A French mathematician known for work in algebra, number theory, and the theory of equations. Hermite proved that e is transcendental, a landmark precursor to the theorem.
Squaring the circle: The problem of constructing a square equal in area to a given circle using only straightedge and compass. The transcendence of π proves this classical construction impossible.
Baker's theorem: A theorem giving strong lower bounds for nonzero linear forms in logarithms of algebraic numbers. It develops quantitative methods in the same broader study of transcendental values.
Linear independence: A set of vectors is linearly independent when only the trivial linear combination equals zero. The theorem rules out every nontrivial algebraic-coefficient relation among the specified exponentials.
Polynomial: An expression formed from variables and coefficients using addition, subtraction, and multiplication with nonnegative integer powers. Algebraicity is defined by being a root of a polynomial over the rationals.
Ferdinand von Lindemann: A German mathematician whose work included proving the transcendence of π. Lindemann extended Hermite's methods to establish π's transcendence.
Straightedge and compass construction: A geometric construction using only an unmarked straightedge and a compass. Such constructions produce only algebraic lengths, so π's transcendence blocks circle squaring.
Lindemann–Weierstrass theorem for logarithms: The study of linear combinations of logarithms of algebraic numbers and their arithmetic properties. Taking logarithms connects exponential independence to questions about relations among logarithms.

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