symmetric monoidal (∞,1)-category of spectra
transfinite arithmetic, cardinal arithmetic, ordinal arithmetic
prime field, p-adic integer, p-adic rational number, p-adic complex number
arithmetic geometry, function field analogy
Algebraic number theory studies algebraic numbers, number fields and related algebraic structures.
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 $k$ a (algebraic) number field $K = k[P]$ over $k$ is the minimal field containing all the roots of a given polynomial $P$ with coefficients in $k$. 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 $n$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)
separable extension, normal extension of fields?, Galois group, Galois extension, abelian extension of fields?, cyclotomic field
Brower group?, Galois cohomology
Dedekind ring, principal ideal ring?, unique factorization domain, integral closure
discriminant, resultant, Euclid algorithm?
Diophantine equation, Matiyasevich theorem?
discrete valuation, valuation ring, valuation ideal?, archimedean valuation
conductor, ideal class group, Picard group, Milnor K-group?
class field theory, global class field theory?, local class field theory?, higher class field theory?, Hilbert class field?
reciprocity law, Artin reciprocity law, Weil reciprocity law
arithmetic scheme, Arakelov geometry, field with one element
L-function, motive, Riemann conjecture, algebraic K-theory, Grothendieck Galois theory
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.
Jürgen Neukirch, Algebraische Zahlentheorie (1992), English translation Algebraic Number Theory, Grundlehren der Mathematischen Wissenschaften 322, 1999 (pdf)
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
See also