definition of integers modulo prime number field
Topics
About: field
The table of contents of this article
Starting Context
- The reader knows a definition of integers modulo natural number ring.
- The reader knows a definition of field.
Target Context
- The reader will have a definition of integers modulo prime number field.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( p\): \(\in \{\text{ the prime numbers }\}\)
\(*\mathbb{Z} / p\): \(= \text{ the integers modulo natural number ring }\), \(\in \{\text{ the fields }\}\)
//
Conditions:
//
Frequently, it is denoted also as \(\mathbb{F}_p\): \(\mathbb{Z} / p = \mathbb{F}_p\).
2: Note
\(\mathbb{Z} / p\) is indeed a field, by the proposition that the quotient ring of the integers ring by any prime principal ideal is a field.
So, when \(n\) is any prime number, the integers modulo natural number ring, \(\mathbb{Z} / n\), inevitably becomes a field
