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Hilbert unified the field of algebraic number theory with his 1897 treatise Zahlbericht (literally "report on numbers"). He also resolved a significant number-theory problem formulated by Waring in 1770.
Corry (1996) and Schappacher (2005) and the English introduction to (Hilbert 1998) give detailed discussions of the history and influence of Hilbert's Zahlbericht. Some earlier reports on number theory include the report by H. J. S. Smith in 6 parts between 1859 and 1865, reprinted in Smith (1965), and the report by Brill & Noether (1894).
The following are the headers for Hilbert's 23 problems as they appeared in the 1902 translation in the Bulletin of the American Mathematical Society. [1] 1. Cantor's problem of the cardinal number of the continuum. 2. The compatibility of the arithmetical axioms. 3. The equality of the volumes of two tetrahedra of equal bases and equal ...
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers , finite fields , and function fields .
The Gelfond–Schneider constant or Hilbert number [1] is two to the power of the square root of two: . 2 √ 2 ≈ 2.665 144 142 690 225 188 650 297 249 8731.... which was proved to be a transcendental number by Rodion Kuzmin in 1930. [2]
Number theory is a branch of pure ... Other than a treatise on squares in arithmetic progression by ... in 1970, it was proven, as a solution to Hilbert's ...
The local-global principle says that a general result about a rational number or even all rational numbers can often be established by verifying that the result holds true for each of the p-adic number systems. There is also more recent work on Hilbert's eleventh problem studying when an integer can be represented by a quadratic form.
Translated by W. Ewald as 'The Grounding of Elementary Number Theory', pp. 266–273 in Mancosu (ed., 1998) From Brouwer to Hilbert: The debate on the foundations of mathematics in the 1920s, Oxford University Press. New York. S.G. Simpson, 1988. Partial realizations of Hilbert's program (pdf). Journal of Symbolic Logic 53:349–363. R. Zach ...