symmetric monoidal (∞,1)-category of spectra
An algebraic number is a root of a polynomial equation with integer coefficients (or, equivalently with rational coeffients). An algebraic integer is a root of a monic polynomial with integer coefficients. Given a field a (algebraic) number field over is the minimal field containing all the roots of a given polynomial with coefficients in . Usually one considers algebraic number fields over rational numbers.
The main direction in algebraic number theory is the class field theory which roughly studies finite abelian extensions of number fields. The one dimensional class field theory stems from the ideas of Kronecker and Weber, and results of Hilbert soon after them. Main results of the theory belong to the first half of the 20th century (Hilbert, Artin, Tate, Hasse…) and are quite different for the local field from the global field case. Generalizations for higher dimensional fields came later under now active higher class field theory, which is usually formulated in terms of algebraic K-theory and is closely related to deep questions of algebraic geometry (Tate, Kato, Saito etc.).
The circle of Lab entries belonging or related closely to algebraic number theory is in its infancy, and the partial list of entries some of which are started and most of which are to be created should include (the entries grouped by similarity)
Albrecht Fröhlich, J. W. S. Cassels (editors), Algebraic number theory, Acad. Press 1967, with many reprints; Fröhlich, Cassels, Birch, Atiyah, Wall, Gruenberg, Serre, Tate, Heilbronn, Rouqette, Kneser, Hasse, Swinerton-Dyer, Hoechsmann, systematic lecture notes from the instructional conference at Univ. of Sussex, Brighton, Sep. 1-17, 1965.
Albrecht Fröhlich, Martin J. Taylor, Algebraic number theory, Cambridge Studies in Advanced Mathematics 27, 1993
The following survey of Connes-Marcolli work has an accessible quick introduction to algebraic number theory