Hilbert's seventh problem: Difference between revisions
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==Statement of the problem== | ==Statement of the problem== | ||
Two specific equivalent<ref>{{cite book|first=N. I. | | Two specific equivalent questions are asked:<ref>{{cite book | ||
| last = Feldman | first = N. I. | author-link = Naum Il'ich Feldman | |||
| last2 = Nesterenko | first2 = Yu. V. | |||
| editor-last1 = Parshin | editor-first1 = A. N. | |||
| editor-last2 = Shafarevich | editor-first2 = I. R. | |||
| series = Number Theory IV | |||
| title = Transcendental Numbers | |||
| url = https://archive.org/details/numbertheorytran00pars | |||
| url-access = limited | |||
| date = 1998 | |||
| publisher = Springer-Verlag Berlin Heidelberg | |||
| isbn = 978-3-540-61467-8 | |||
| pages = [https://archive.org/details/numbertheorytran00pars/page/n145 146]–147 | |||
}}</ref> | |||
#In an [[isosceles triangle]], if the ratio of the base [[angle]] to the angle at the vertex is [[algebraic number|algebraic]] but [[irrational number|not rational]], is then the ratio between base and side always [[transcendental number|transcendental]]? | #In an [[isosceles triangle]], if the ratio of the base [[angle]] to the angle at the vertex is [[algebraic number|algebraic]] but [[irrational number|not rational]], is then the ratio between base and side always [[transcendental number|transcendental]]? | ||
#Is <math>a^b</math> always [[transcendental number|transcendental]], for [[algebraic number|algebraic]] <math>a \not\in \{0,1\}</math> and [[irrational number|irrational]] algebraic <math>b</math>? | #Is <math>a^b</math> always [[transcendental number|transcendental]], for [[algebraic number|algebraic]] <math>a \not\in \{0,1\}</math> and [[irrational number|irrational]] algebraic <math>b</math>? | ||
Latest revision as of 03:02, 29 December 2025
Template:Short description Hilbert's seventh problem is one of David Hilbert's list of open mathematical problems posed in 1900. It concerns the irrationality and transcendence of certain numbers (Irrationalität und Transzendenz bestimmter Zahlen).
Statement of the problem
Two specific equivalent questions are asked:[1]
- In an isosceles triangle, if the ratio of the base angle to the angle at the vertex is algebraic but not rational, is then the ratio between base and side always transcendental?
- Is always transcendental, for algebraic and irrational algebraic ?
Solution
The question (in the second form) was answered in the affirmative by Aleksandr Gelfond in 1934, and refined by Theodor Schneider in 1935. This result is known as Gelfond's theorem or the Gelfond–Schneider theorem. (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 generalizations, this is the case
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. Baker was awarded a Fields Medal in 1970 for this achievement.
See also
References
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Bibliography
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External links
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