Kronecker–Weber theorem

In algebraic number theory, it can be shown that every cyclotomic field is an abelian extension of the rational number field Q, having Galois group of the form ${\displaystyle (\mathbb {Z} /n\mathbb {Z} )^{\times }}$. The Kronecker–Weber theorem provides a partial converse: every finite abelian extension of Q is contained within some cyclotomic field. In other words, every algebraic integer whose Galois group is abelian can be expressed as a sum of roots of unity with rational coefficients. For example,

${\displaystyle {\sqrt {5}}=e^{2\pi i/5}-e^{4\pi i/5}-e^{6\pi i/5}+e^{8\pi i/5},}$ ${\displaystyle {\sqrt {-3}}=e^{2\pi i/3}-e^{4\pi i/3},}$ and ${\displaystyle {\sqrt {3}}=e^{\pi i/6}-e^{5\pi i/6}.}$

The theorem is named after Leopold Kronecker and Heinrich Martin Weber.

Field-theoretic formulation

The Kronecker–Weber theorem can be stated in terms of fields and field extensions. Precisely, the Kronecker–Weber theorem states: every finite abelian extension of the rational numbers Q is a subfield of a cyclotomic field. That is, whenever an algebraic number field has a Galois group over Q that is an abelian group, the field is a subfield of a field obtained by adjoining a root of unity to the rational numbers.

For a given abelian extension K of Q there is a minimal cyclotomic field that contains it. The theorem allows one to define the conductor of K as the smallest integer n such that K lies inside the field generated by the n-th roots of unity. For example the quadratic fields have as conductor the absolute value of their discriminant, a fact generalised in class field theory.

History

The theorem was first stated by Kronecker (1853) though his argument was not complete for extensions of degree a power of 2. Weber (1886) published a proof, but this had some gaps and errors that were pointed out and corrected by Neumann (1981). The first complete proof was given by Hilbert (1896).

Generalizations

Lubin and Tate (1965, 1966) proved the local Kronecker–Weber theorem which states that any abelian extension of a local field can be constructed using cyclotomic extensions and Lubin–Tate extensions. Hazewinkel (1975), Rosen (1981) and Lubin (1981) gave other proofs.

Hilbert's twelfth problem asks for generalizations of the Kronecker–Weber theorem to base fields other than the rational numbers, and asks for the analogues of the roots of unity for those fields. A different approach to abelian extensions is given by class field theory.

References

• Ghate, Eknath (2000), "The Kronecker-Weber theorem" (PDF), in Adhikari, S. D.; Katre, S. A.; Thakur, Dinesh (eds.), Cyclotomic fields and related topics (Pune, 1999), Bhaskaracharya Pratishthana, Pune, pp. 135–146, MR 1802379
• Greenberg, M. J. (1974). "An Elementary Proof of the Kronecker-Weber Theorem". American Mathematical Monthly. 81 (6): 601–607. doi:10.2307/2319208. JSTOR 2319208.
• Hazewinkel, Michiel (1975), "Local class field theory is easy" (PDF), Advances in Mathematics, 18 (2): 148–181, doi:10.1016/0001-8708(75)90156-5, ISSN 0001-8708, MR 0389858
• Hilbert, David (1896), "Ein neuer Beweis des Kronecker'schen Fundamentalsatzes über Abel'sche Zahlkörper.", Nachrichten der Gesellschaft der Wissenschaften zu Göttingen (in German): 29–39
• Kronecker, Leopold (1853), "Über die algebraisch auflösbaren Gleichungen", Berlin K. Akad. Wiss. (in German): 365–374, ISBN 9780821849828, Collected works volume 4
• Kronecker, Leopold (1877), "Über Abelsche Gleichungen", Berlin K. Akad. Wiss. (in German): 845–851, ISBN 9780821849828, Collected works volume 4
• Lemmermeyer, Franz (2005), "Kronecker-Weber via Stickelberger", Journal de théorie des nombres de Bordeaux, 17 (2): 555–558, arXiv:1108.5671, doi:10.5802/jtnb.507, ISSN 1246-7405, MR 2211307
• Lubin, Jonathan (1981), "The local Kronecker-Weber theorem", Transactions of the American Mathematical Society, 267 (1): 133–138, doi:10.2307/1998574, ISSN 0002-9947, JSTOR 1998574, MR 0621978
• Lubin, Jonathan; Tate, John (1965), "Formal complex multiplication in local fields", Annals of Mathematics, Second Series, 81 (2): 380–387, doi:10.2307/1970622, ISSN 0003-486X, JSTOR 1970622, MR 0172878
• Lubin, Jonathan; Tate, John (1966), "Formal moduli for one-parameter formal Lie groups", Bulletin de la Société Mathématique de France, 94: 49–59, doi:10.24033/bsmf.1633, ISSN 0037-9484, MR 0238854
• Neumann, Olaf (1981), "Two proofs of the Kronecker-Weber theorem "according to Kronecker, and Weber"", Journal für die reine und angewandte Mathematik, 323 (323): 105–126, doi:10.1515/crll.1981.323.105, ISSN 0075-4102, MR 0611446
• Rosen, Michael (1981), "An elementary proof of the local Kronecker-Weber theorem", Transactions of the American Mathematical Society, 265 (2): 599–605, doi:10.2307/1999753, ISSN 0002-9947, JSTOR 1999753, MR 0610968
• Šafarevič, I. R. (1951), A new proof of the Kronecker-Weber theorem, Trudy Mat. Inst. Steklov. (in Russian), vol. 38, Moscow: Izdat. Akad. Nauk SSSR, pp. 382–387, MR 0049233
• Schappacher, Norbert (1998), "On the history of Hilbert's twelfth problem: a comedy of errors", Matériaux pour l'histoire des mathématiques au XX^e siècle (Nice, 1996), Sémin. Congr., vol. 3, Paris: Société Mathématique de France, pp. 243–273, ISBN 978-2-85629-065-1, MR 1640262
• Weber, H. (1886), "Theorie der Abel'schen Zahlkörper", Acta Mathematica (in German), 8: 193–263, doi:10.1007/BF02417089, ISSN 0001-5962

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