Locally compact fields
Let F be a finite field (commutative or not) with the unit-element 1. Its characteristic must clearly be a prime p > l, and the prime ring in F is isomorphic to the prime field F p =Z/p Z, with which we may identify it. Then F may be regarded as a vector-space over F p ; as such, it has an obviously finite dimension ƒ, and the number of its elements is q=p f . If F is a subfield of a field F′; with q′=p f ′ elements, F&#x 2032; may also be regarded e.g. as a left vector-space over F; if its dimension as such is d, we have f′ = df and q′ = q d =p df .
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