# Hilbert’s Seventh Problem

Irrationality and Transcendence of Certain Numbers: Is ab transcendental, for algebraic a ≠ 0,1 and irrational algebraic b

A number is called algebraic if it can be the zero of a polynomial with rational coefficients. For example, 2 is a zero of the polynomial x − 2, and √2 is a zero of the polynomial x2 − 2. Algebraic numbers can be either rational or irrational; transcendental numbers like π are irrational numbers that are not algebraic. Hilbert’s seventh problem concerns powers of algebraic numbers. Consider the expression ab, where a is an algebraic number other than 0 or 1 and b is an irrational algebraic number. Must ab be transcendental? In 1934, Aleksandr Gelfond showed that the answer is yes.

## Hilbert’s Twenty-second Problem

Hilbert’s Twenty-second Problem: Uniformization of analytic relations by means of automorphic functions Hilbert’s 22nd problem asks whether every algebraic or analytic curve — solutions to polynomial equations — can be written…

## Hilbert’s Twenty-first Problem

Hilbert’s Twenty-first Problem: Proof of the existence of linear differential equations having a prescribed monodromic group Hilbert’s 21st problem is about the existence of certain systems of differential equations with given singular points…

## Hilbert’s Twentieth Problem

Hilbert’s Twentieth Problem: Do all variational problems with certain boundary conditions have solutions Do solutions in general exist? The calculus of variations is a field concerned with optimizing certain types of functions called functionals.…

## Hilbert’s Problem #19

Are The Solutions of Regular Problems in The Calculus of Variations Always Necessarily Analytic: Are the solutions of regular problems in the calculus of variations always necessarily analytic Do solutions in general exist?…

## Hilbert’s Eighteenth Problem

Hilbert’s Eighteenth Problem: (a) Is there a polyhedron that admits only an anisohedral tiling in three dimensions? (b) What is the densest sphere packing? Hilbert’s 18th problem is a collection of several questions…

## Hilbert’s Seventh Problem

EHilbert’s Seventh Problem: Express a nonnegative rational function as quotient of sums of squares Some polynomials with inputs in the real numbers always take non-negative values; an easy example is x2 + y2. Hilbert’s 17th problem asks…

## Hilbert’s Sixteenth Problem

ToHilbert’s Sixteenth Problem: Describe relative positions of ovals originating from a real algebraic curve and as limit cycles of a polynomial vector field on the plane. Hilbert’s 16th problem is an expansion of grade school graphing questions.…

## Hilbert’s Fifteenth Problem

Hilbert’s Fifteenth Problem is the igorous foundation of Schubert’s enumerative calculus. Hilbert’s 15th problem is another question of rigor. He called for mathematicians to put Schubert’s enumerative calculus, a branch of…