Hilbert's seventh problem is one of David Hilbert

David Hilbert

David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...

's list of open mathematical problems posed in 1900. It concerns the irrationality

Irrational number

In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....

 and transcendence

Transcendental number

In mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...

 of certain numbers (Irrationalität und Transzendenz bestimmter Zahlen). Two specific questions are asked:

1. In an isosceles triangle, if the ratio of the base angle
   Angle
   In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...
   
    to the angle at the vertex is algebraic
   Algebraic number
   In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients. Numbers such as π that are not algebraic are said to be transcendental; almost all real numbers are transcendental...
   
    but not rational
   Irrational number
   In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....
   
   , is then the ratio between base and side always transcendental
   Transcendental number
   In mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...
   
   ?
2. Is always transcendental
   Transcendental number
   In mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...
   
   , for algebraic
   Algebraic number
   In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients. Numbers such as π that are not algebraic are said to be transcendental; almost all real numbers are transcendental...
   
     and irrational
   Irrational number
   In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....
   
    algebraic ?

The second question was answered in the affirmative by Aleksandr Gelfond in 1934, and refined by Theodor Schneider

Theodor Schneider

Theodor Schneider was a German mathematician, best known for providing proof of what is now known as the Gelfond–Schneider theorem in 1935....

 in 1935. This result is known as Gelfond's theorem or the Gelfond–Schneider theorem

Gelfond–Schneider theorem

In mathematics, the Gelfond–Schneider theorem establishes the transcendence of a large class of numbers. It was originally proved independently in 1934 by Aleksandr Gelfond and Theodor Schneider...

. (The restriction to irrational b is important, since it is easy to see that is algebraic for algebraic a and rational b.)

From the point of view of generalisations, this is the case


of the general linear form in logarithms which was attacked by
Gelfond and then solved by Alan Baker.
It is called the Gelfond conjucture or Baker's theorem

Baker's theorem

In transcendence theory, a mathematical discipline, Baker's theorem gives a lower bound for linear combinations of logarithms of algebraic numbers...

. Baker was rewarded
as a Fields Medal winner in 1970 because of this.

The first question is a consequence of the second question.

External links

• English translation of Hilbert's original address