Field has Prime Characteristic p iff exists Monomorphism from Field of Integers Modulo p

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Theorem

Let $F$ be a field.

Then:

there exists some prime number $p$ such that $\Char F = p$

if and only if:

there exists a field monomorphism $\phi: \Z_p \to F$

where:

$\Char F$ denotes the characteristic of $F$.
$\Z_p$ denotes the field of integers modulo $p$.


Proof

Let $\struct {F, +, \times}$ be a field whose zero is $0$ and whose unity is $1$.


Sufficient Condition

Let there exists some prime number $p$ such that $\Char F = p$.


Let us define the mapping $\phi: \Z_p \to F$ as:

$\forall \eqclass n p \in \Z_p: \map \phi {\eqclass n p} = n \cdot 1$

where $n \cdot 1$ denotes the power of $1$ in the context of the additive group $\struct {F, +}$:

$n \cdot 1 = \begin {cases}

0 & : n = 0 \\ \paren {\paren {n - 1} \cdot 1} + 1 & : n > 0 \end {cases}$


We show that $\phi$ is the field monomorphism required.


By definition, a field monomorphism is a field homomorphism which is also an injection.


We have:

\(\ds \map \phi {\eqclass a p} + \map \phi {\eqclass b p}\) \(=\) \(\ds \paren {a \cdot 1} + \paren {b \cdot 1}\)
\(\ds \) \(=\) \(\ds \paren {a + b} \cdot 1\) Powers of Group Elements: Additive Notation
\(\ds \) \(=\) \(\ds \map \phi {\eqclass {a + b} p}\) Definition of $\phi$


\(\ds \map \phi {\eqclass a p} \times \map \phi {\eqclass b p}\) \(=\) \(\ds \paren {a \cdot 1} \times \paren {b \cdot 1}\) Definition of $\phi$
\(\ds \) \(=\) \(\ds \paren {a \times b} \cdot \paren {1 \times 1}\) Product of Integral Multiples
\(\ds \) \(=\) \(\ds \paren {a \times b} \cdot 1\) Definition of Multiplicative Identity
\(\ds \) \(=\) \(\ds \map \phi {\eqclass {a \times b} p}\) Definition of $\phi$

Then by definition $\phi$ is a (field) homomorphism.


It remains to be shown that $\phi$ is an injection.

Let $\eqclass a p, \eqclass b p \in \Z_p$ such that:

$\phi {\eqclass a p} = \phi {\eqclass b p}$


This needs considerable tedious hard slog to complete it.
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Necessary Condition

Let there exist a field monomorphism $\phi: \Z_p \to F$.

We require to show that $\Char F = p$.

It is sufficient to show that:

$\forall a \in F: n \circ a = 0 \iff n = p$


This needs considerable tedious hard slog to complete it.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Finish}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.



Sources

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