968: Primitive n-th Root of 1 on Field

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definition of primitive n-th root of 1 on field

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Topics

About: field

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Note

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Starting Context

• The reader knows a definition of field.

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Target Context

• The reader will have a definition of primitive n-th root of 1 on field.

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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.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$F$: $\in \{\text{ the fields }\}$
$n$: $\in \mathbb{N}\setminus \{0\}$
$\ast {\omega }_n$: $\in F$
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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 $(-1{)}^1=-1\ne 1$.

For example, when $F=\mathbb{C}$, there is the primitive 3rd roots of 1 on $F$: $(e^{i2\pi /3}{)}^3=1$ and $(e^{i2\pi /3}{)}^1\ne 1$ and $(e^{i2\pi /3}{)}^2\ne 1$; $(e^{i2\pi 2/3}{)}^3=1$ and $(e^{i2\pi 2/3}{)}^1\ne 1$ and $(e^{i2\pi 2/3}{)}^2\ne 1$.

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References

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