Tate algebra

In rigid analysis, a branch of mathematics, the Tate algebra over a complete ultrametric field k, named for John Tate, is the subring R of the formal power series ring k [ [ t 1 , . . . , t n ] ] consisting of ∑ a I t I such that | a I | → 0 as I → ∞ . The maximal spectrum of R is then a rigid-analytic space.

Define the Gauss norm of f = ∑ a I t I in R by

This makes R a Banach k-algebra.

With this norm, any ideal I of T n is closed and T n / I is a finite field extension of ground field K .