Field of Prime Characteristic has Unique Prime Subfield

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Theorem

Let $F$ be a field whose characteristic is $p$.

Then there exists a unique $P \subseteq F$ such that:

$(1): \quad P$ is a subfield of $F$
$(2): \quad P \cong \Z_p$.

That is, $P \cong \Z_p$ is a unique minimal subfield of $F$, and all other subfields of $F$ contain $P$.


This field $P$ is called the prime subfield of $F$.


Proof

Let $\struct {F, +, \times}$ be a field whose unity is $1_F$ such that $\Char F = p$.

Let $P$ be a prime subfield of $F$.

From Field has Prime Subfield, this has been shown to exist.


We can consistently define a mapping $\phi: \Z_p \to F$ by:

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


Suppose $a, b \in \eqclass n p$.

Then:

$a = n + k_1 p, b = n + k_2 p$

So:

\(\ds \map \phi a\) \(=\) \(\ds \map \phi {n + k_1 p}\)
\(\ds \) \(=\) \(\ds \paren {n + k_1 p} \cdot 1_F\)
\(\ds \) \(=\) \(\ds n \cdot 1_F + k_1 p \cdot 1_F\)
\(\ds \) \(=\) \(\ds n \cdot 1_F\)

and similarly for $b$, showing that $\phi$ is well-defined.


Let $C_a, C_b \in \Z_p$.

Let $a \in C_a, b \in C_b$ such that $a = a' + k_a p, b = b' + k_b p$.

Then:

\(\ds \map \phi {C_a} + \map \phi {C_b}\) \(=\) \(\ds \map \phi {a' + k_a p} + \map \phi {b' + k_b p}\)
\(\ds \) \(=\) \(\ds \paren {a' + k_a p} \cdot 1_F + \paren {b' + k_b p} \cdot 1_F\)
\(\ds \) \(=\) \(\ds \paren {a' + k_a p + b' + k_b p} \cdot 1_F\) Integral Multiple Distributes over Ring Addition
\(\ds \) \(=\) \(\ds \paren {a' + b'} \cdot 1_F + \paren {\paren {k_a p + k_b p} \cdot 1_F}\)
\(\ds \) \(=\) \(\ds \paren {a' + b'} \cdot 1_F\)
\(\ds \) \(=\) \(\ds \map \phi {C_a +_p C_b}\)

Similarly for $\map \phi {C_a} \times \map \phi {C_b}$.


So $\phi$ is a ring homomorphism.

From Ring Homomorphism from Field is Monomorphism or Zero Homomorphism, it follows that $\phi$ is a ring monomorphism.


Thus it follows that $P = \Img \phi$ is a subfield of $F$ such that $P \cong \Z_p$.


Let $K$ be a subfield of $F$.

let $P = \Img \phi$ as defined above.

We know that $1_F \in K$.

It follows that $1_F \in K \implies P \subseteq K$.

Thus $K$ contains a subfield $P$ such that $P$ is isomorphic to $\Z_p$.


The uniqueness of $P$ follows from the fact that if $P_1$ and $P_2$ are both minimal subfields of $F$, then $P_1 \subseteq P_2$ and $P_2 \subseteq P_1$, thus $P_1 = P_2$.

$\blacksquare$


Sources

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