Hilbert's seventh problem is one of David Hilbert's list of open mathematical problems posed in 1900. It concerns the

irrationality Irrationality is cognition, thinking, talking, or acting without inclusion of rationality. It is more specifically described as an action or opinion given through inadequate use of reason, or through emotional distress or cognitive deficiency. ...

and

transcendence Transcendence, transcendent, or transcendental may refer to: Mathematics * Transcendental number, a number that is not the root of any polynomial with rational coefficients * Algebraic element or transcendental element, an element of a field exten ...

of certain numbers (''Irrationalität und Transzendenz bestimmter Zahlen'').

Statement of the problem

Two specific equivalent questions are asked: #In an

isosceles triangle In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...

, if the ratio of the base

angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles ...

to the angle at the vertex is algebraic but not rational, is then the ratio between base and side always

transcendental Transcendence, transcendent, or transcendental may refer to: Mathematics * Transcendental number, a number that is not the root of any polynomial with rational coefficients * Algebraic element or transcendental element, an element of a field exten ...

? #Is

a^b

always

, for algebraic

a \not\in \

and

irrational Irrationality is cognition, thinking, talking, or acting without inclusion of rationality. It is more specifically described as an action or opinion given through inadequate use of reason, or through emotional distress or cognitive deficiency. ...

algebraic

b

Solution

The question (in the second form) was answered in the affirmative by Aleksandr Gelfond in 1934, and refined by

Theodor Schneider __NOTOC__ Theodor Schneider (7 May 1911, Frankfurt am Main – 31 October 1988, Freiburg im Breisgau) was a German mathematician, best known for providing proof of what is now known as the Gelfond–Schneider theorem. Schneider studied from 19 ...

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

Gelfond–Schneider theorem In mathematics, the Gelfond–Schneider theorem establishes the transcendence of a large class of numbers. History It was originally proved independently in 1934 by Aleksandr Gelfond and Theodor Schneider. Statement : If ''a'' and ''b'' ar ...

. (The restriction to irrational ''b'' is important, since it is easy to see that

a^b

is algebraic for algebraic ''a'' and rational ''b''.) From the point of view of generalizations, this is the case :

b \ln + \ln = 0

of the general linear form in logarithms which was studied by Gelfond and then solved by Alan Baker. It is called the Gelfond conjecture or

Baker's theorem In transcendental number theory, a mathematical discipline, Baker's theorem gives a lower bound for the absolute value of linear combinations of logarithms of algebraic numbers. The result, proved by , subsumed many earlier results in transcendenta ...

. Baker was awarded a Fields Medal in 1970 for this achievement.

References

Bibliography

* *

External links

English translation of Hilbert's original address
{{Authority control #07

Statement of the problem

Solution

See also

References

Bibliography

External links