Abstract
In this chapter one finds a presentation of the work of Hilbert, who in his report on algebraic numbers summarized the state of their theory at the end of the nineteenth century, as well of Hensel, who created p-adic and \(\mathfrak p\)-adic numbers, which turned out to be an indispensable tool in future research. In the last part of the chapter the first steps towards creation of the class-field theory, characterizing Abelian extensions of algebraic number fields, are described.
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Notes
- 1.
Serge Lang (1927–2005), professor at Columbia University and at Yale. See [2075].
- 2.
Martin Eichler (1912–1992), professor in Münster, Marburg and Basel. See [2186].
- 3.
Abraham Robinson (1918–1974), professor in Toronto at the Hebrew University, UCLA, and Yale. See [2681].
- 4.
Speiser formulated to in a more elementary but equivalent way.
- 5.
Emmy Noether (1882–1935), daughter of Max Noether, worked in Göttingen and Bryn Mawr. See [929, 4137].
- 6.
Nathan Jacobson (1910–1999), professor at the University of North Carolina, John Hopkins University and Yale. See [289].
- 7.
This way of writing has been first used in Kronecker’s thesis [2276].
- 8.
Ivan Morton Niven (1915–1999), professor at the University of Illinois, Purdue University and the University of Oregon.
- 9.
For the history of this notion see K. Conrad [763].
- 10.
“I was led for the first time to the notion of a general group determinant during my study of discriminants of a general normal field \(\varOmega \), when I considered such (very useful) bases on \(\varOmega \) which consist of conjugates to a single number \(\omega \) (sometimes the system of all integers of \(\varOmega \) has such a basis, e.g. when \(\omega \) is an mth root if unity and m is not divisible by a square, and the same applies also to all subfields of \(\varOmega \), e.g. to all quadratic fields with an odd fundamental number”.
- 11.
Anders Wiman (1865–1959), professor in Uppsala. See [3029].
- 12.
Vera Myller-Lebedeff (1880–1970), professor in Iaşi.
- 13.
Nesmith Corbett Ankeny (1927–1993), professor at M.I.T.
- 14.
Sarvadaman Chowla (1907–1995), professor in Delhi, Benares, Waltair, Lahore, at the University of Kansas, University of Colorado and Pennsylvania State University. See [159].
- 15.
“The proof of this theorem presents considerable difficulties; it depends essentially on the theory of relative quadratic extensions”.
- 16.
Eustachy Żyliński (1889–1951), professor in Lwów and Gliwice.
- 17.
Marshall Hall Jr. (1910–1990), professor at the Ohio State University and CalTech.
- 18.
Peter Arthur Barry Pleasants (1939–2008), lecturer in Cardiff, the University of New England, Macquarie University and the University of the South Pacific.
- 19.
Gustav Eric Wahlin (1880–1948), professor at the University of Missouri.
- 20.
In a later paper Hensel [1779] tried, rather unsuccessfully, to reconcile the usual convergence of series with \(\mathfrak p\)-adical convergence.
- 21.
Hensel’s proof gives actually the unique factorization property for polynomials in one variable over an arbitrary field.
- 22.
Gustave Dumas (1872–1955), professor in Zürich and Lausanne. See [896].
- 23.
The name “p-adic numbers” seems to appear for the first time in Hensel’s paper [1784], in which he considered quadratic extensions of \(\mathbf{Q}_p\).
- 24.
Kurt Mahler (1903–1988), professor in Manchester and Canberra. See [405, 598, 725, 4128].
- 25.
It was planned to have two volumes, but only the first volume appeared.
- 26.
[1780], p. VII.
- 27.
Tonio Rella (1888–1945), professor in Wien and Graz.
- 28.
Ernst Steinitz (1871–1928), professor in Breslau and Kiel. See [3507].
- 29.
Adolf Fraenkel (1801–1965), professor in Marburg, Kiel and Jerusalem.
- 30.
Roquette wrote in [3502] that “Fraenkel’s paper is completely forgotten today”.
- 31.
József Kürschak (1864–1933), professor at the Technical University in Budapest
- 32.
Georg Cantor (1845–1918), professor in Halle. See [3346].
- 33.
Alexander Ostrowski (1893–1986), professor in Basel. See [1061, 2045].
- 34.
Wolfgang Krull (1899–1971), professor in Freiburg, Erlangen and Bonn. See [3078, 3663].
- 35.
Otto Schilling (1911–1973), professor at the Purdue University.
- 36.
Otto Endler (1929–1988), professor at the IMPA in Rio de Janeiro.
- 37.
Reinhold Strassmann (1893–1944), worked in an insurance company in Marburg. Killed in Auschwitz.
- 38.
Max Deuring (1907–1984), professor in Jena, Marburg, Hamburg and Göttingen. See [1060, 3501].
- 39.
Kronecker’s Youth Dream.
- 40.
Teiji Takagi (1875–1960), professor in Tokyo. See [1891, 1996, 2897].
- 41.
Christoph Gudermann (1798–1852), professor in Münster, teacher of Weierstrass.
- 42.
Edmund Taylor Whittaker (1873–1956), professor in Dublin and Edinburgh. See [2795].
- 43.
George Neville Watson (1886–1965), professor in Birmingham. See [3383].
- 44.
Another proof of the last assertion has been given later by Eichler [1058].
- 45.
Georg Pick (1859–1942), professor in Prague.
- 46.
Armand Borel (1923–2003), professor in Zürich and at IAS in Princeton. See [114].
- 47.
Carl Samuel Herz (1930–1995), professor at the Cornell University and McGill University. See [1].
- 48.
Cyclic groups and fields were called regular by Weber.
- 49.
He pointed out that a similar result under somewhat different assumptions appears on p. 62 in the book [2874] of Minkowski.
- 50.
Aurel Friedrich Wintner (1903–1958), professor in Baltimore. See [1636].
- 51.
Note that in these papers as well as in his thesis ([1296], p. 5) Fueter used the name “Strahl” (i.e. “ray”) for any multiplicative group consisting of numbers.
- 52.
“Because of the close relation, which according to theorem 94, exists between the field K and certain ideal classes on k, the field K will be called the class-field of the field k”.
- 53.
“We present the following theorems, which in certain special cases were established in the preceding, but whose complete proofs, I am sure, could be obtained on the basis of methods showed by me.”
- 54.
Olga Taussky-Todd (1906–1995), wife of John Todd, worked at the National Bureau of Standards and CalTech. See [1620, 1873, 2162].
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Narkiewicz, W. (2018). The Turn of the Century. In: The Story of Algebraic Numbers in the First Half of the 20th Century. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-03754-3_2
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