definition of primitive n-th root of 1 on field
Topics
About: field
The table of contents of this article
Starting Context
- The reader knows a definition of field.
Target Context
- The reader will have a definition of primitive n-th root of 1 on 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:
\( F\): \(\in \{\text{ the fields }\}\)
\( n\): \(\in \mathbb{N} \setminus \{0\}\)
\(*\omega_n\): \(\in F\)
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Conditions:
\(n\) is the smallest \(j \in \mathbb{N} \setminus \{0\}\) such that \(\omega_n^j = 1\)
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2: Note
There may not be any primitive n-th root of 1 on \(F\).
There may be some multiple primitive n-th roots of 1 on \(F\).
For example, when \(F = \mathbb{R}\), there is no primitive 3rd root of 1 on \(F\): \(1^3 = 1\) but \(1^1 = 1\), so, \(1\) is not any primitive 3rd root of 1 on \(F\), while there is the primitive 2nd root of \(1\) on \(F\): \((-1)^2 = 1\) and \((-1)^1 = -1 \neq 1\).
For example, when \(F = \mathbb{C}\), there is the primitive 3rd roots of 1 on \(F\): \((e^{i 2 \pi / 3})^3 = 1\) and \((e^{i 2 \pi / 3})^1 \neq 1\) and \((e^{i 2 \pi / 3})^2 \neq 1\); \((e^{i 2 \pi 2 / 3})^3 = 1\) and \((e^{i 2 \pi 2 / 3})^1 \neq 1\) and \((e^{i 2 \pi 2 / 3})^2 \neq 1\).
