Local Field - Non-archimedean Local Field Theory

Non-archimedean Local Field Theory

For a non-archimedean local field F (with absolute value denoted by |·|), the following objects are important:

• its ring of integers which is a discrete valuation ring, is the closed unit ball of F, and is compact;
• the units in its ring of integers which forms a group and is the unit sphere of F;
• the unique non-zero prime ideal in its ring of integers which is its open unit ball ;
• a generator ϖ of called a uniformizer of F;
• its residue field which is finite (since it is compact and discrete).

Every non-zero element a of F can be written as a = ϖnu with u a unit, and n a unique integer. The normalized valuation of F is the surjective function v : F → Z ∪ {∞} defined by sending a non-zero a to the unique integer n such that a = ϖnu with u a unit, and by sending 0 to ∞. If q is the cardinality of the residue field, the absolute value on F induced by its structure as a local field is given by

An equivalent definition of a non-archimedean local field is that it is a field that is complete with respect to a discrete valuation and whose residue field is finite.