symmetric monoidal (∞,1)-category of spectra
A number field is a finite field extension of the field of rational numbers, , In other words, a field of characteristic zero such that under the field homomorphism , the field is a finite-dimensional vector space over with respect to the scalar multiplication action of
on the underlying additive group of .
the rational numbers ;
the Gaussian numbers ;
for certain the quadratic field? ;
Number fields are the basic objects of study in algebraic number theory. For example, one is typically interested in the arithmetic structure of , including for example the structure of the ring of algebraic integers in , the decomposition of primes in in terms of prime ideals in , the structure of the unit group of , the structure of the ideal class group?, the detailed study of the zeta function of , and much more.
Number fields are examples of global field?s, in fact they are the global fields of characteristic zero. They are often studied in terms of how they embed in their rings of adeles , which are built from the local completions of .