ヒルベルト問題

リスト

1. a. Is there a transfinite number between that of a denumerable set and the numbers of the continuum?
   1. b. Can the continuum of numbers be considered a well ordered set?
2. Can it be proven that the axioms of logic are consistent?
3. Give two tetrahedra which cannot be decomposed into congruent tetrahedra directly or by adjoining congruent tetrahedra.
4. Find geometries whose axioms are closest to those of Euclidean geometry if the ordering and incidence axioms are retained, the congruence axioms weakened, and the equivalent of the parallel postulate omitted.
5. Can the assumption of differentiability for functions defining a continuous transformation group be avoided? (This is a generalization of the Cauchy functional equation.)
6. Can physics be axiomized?
7. Let be algebraic and irrational. Is then transcendental?
8. Prove the Riemann hypothesis.
9. Construct generalizations of the reciprocity theorem of number theory.
10. Does there exist a universal algorithm for solving Diophantine equations?
11. Extend the results obtained for quadratic fields to arbitrary integer algebraic fields.
12. Extend a theorem of Kronecker to arbitrary algebraic fields by explicitly constructing Hilbert class fields using special values.
13. Show the impossibility of solving the general seventh degree equation by functions of two variables.
14. []Show the finiteness of systems of relatively integral functions.
15. Justify Schubert's enumerative geometry
16. Develop a topology of real algebraic curves and surfaces.
17. Find a representation of definite form by squares.
18. Build spaces with congruent polyhedra.
19. Analyze the analytic character of solutions to variational problems.
20. Solve general boundary value problems.
21. Solve differential equations given a monodromy group. More technically, prove that there always exists a Fuchsian system with given singularities and a given monodromy group.
22. Uniformization.
23. Extend the methods of calculus of variations.

1. 連続体問題
2. 算術の公理の無矛盾
3. 底面積と高さの等しい四面体の体積が等しいことを合同定理だけでは証明できないこと
4. 直線が最短距離を与える幾何学の組織的研究
5. 位相群がLie群になるための条件
6. 物理学の各部門の公理的扱い
7. 種種の数の超越性の証明
8. 素数分布の問題
9. 一般相互法則
10. 不定方程式の有理整数解の存在するか否かを有限解の手段で判定すること
11. 一般の代数的数体における2次形式論
12. 類体の構成問題
13. 一般の7次代数方程式は2変数の連続関数の合成によっては解けないこと
14. 与えられた多項式の有理関数のなす環が有限個の多項式として生成されるか
15. 代数幾何学の基礎づけ
16. 代数曲線および曲面の位相の研究
17. 実数体上の定符号の有理関数を平方和として表すこと
18. いくつかの多面体と合同な多面体によって空間をうめつくすこと
19. 正則変分問題の解は常に解析関数であるか
20. 一般境界値問題
21. 与えられたモノドロミー群をもつ線形微分方程式は存在するか
22. 保形関数による解析関数の一意化
23. 変分学の方法の研究の展開

*リスト：リスト::数学関連