Questions tagged [local-field]

In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology. Given such a field, an absolute value can be defined on it. There are two basic types of local fields: those in which the absolute value is Archimedean and those in which it is not. In the first case, one calls the local field an Archimedean local field, in the second case, one calls it a non-Archimedean local field. Local fields arise naturally in number theory as completions of global fields.

While Archimedean local fields are quite well known in mathematics for 250 years and more, the first examples of non-Archimedean local fields, the fields of $p$-adic numbers for positive prime integer $p$, were introduced by Kurt Hensel at the end of the 19th century.

Every local field is isomorphic (as a topological field) to one of the following:

• Archimedean local fields (characteristic zero): the real numbers $\Bbb R$, and the complex numbers $\Bbb C$.
• Non-Archimedean local fields of characteristic zero: finite extensions of the $p$-adic numbers $\Bbb Q_p$ (where $p$ is any prime number).
• Non-Archimedean local fields of characteristic $p$ (for $p$ any given prime number): the field of formal Laurent series $\Bbb F_q((T))$ over a finite field $\Bbb F_q$, where $q$ is a power of $p$.

There is an equivalent definition of non-Archimedean local field: it is a field that is complete with respect to a discrete valuation and whose residue field is finite. In particular, of importance in number theory, classes of local fields show up as the completions of algebraic number fields with respect to their discrete valuation corresponding to one of their maximal ideals. Research papers in modern number theory often consider a more general notion, requiring only that the residue field be perfect of positive characteristic, not necessarily finite.