Power to Characteristic of Field is Monomorphism
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Theorem
Let $F$ be a field whose zero is $0_F$ and whose unity is $1_F$.
Let the characteristic of $F$ be $p$ where $p \ne 0$.
Let $\phi: F \to F$ be the mapping on $F$ defined as:
- $\forall x \in F: \map \phi x = x^p$
Then $\phi$ is a (field) monomorphism.
Proof
Let $a, b \in F$.
First note that:
\(\ds \forall k: 0 < k < p: \, \) | \(\ds \binom p k\) | \(\equiv\) | \(\ds 0\) | \(\ds \pmod p\) | Binomial Coefficient of Prime | |||||||||
\(\ds \leadsto \ \ \) | \(\ds \binom p k\) | \(=\) | \(\ds r p\) | for some $r \in \Z$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \binom p k a^k b^{p - k}\) | \(=\) | \(\ds r p \cdot a^k b^{p - k}\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \binom p k a^k b^{p - k}\) | \(=\) | \(\ds 0_F\) | Characteristic of Field by Annihilator: Prime Characteristic | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \sum_{k \mathop = 1}^{p - 1} \binom p k a^k b^{p - k}\) | \(=\) | \(\ds 0_F\) |
So:
\(\ds \map \phi {a + b}\) | \(=\) | \(\ds \paren {a + b}^p\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds a^p + \sum_{k \mathop = 1}^{p - 1} \binom p k a^k b^{p - k} + b^p\) | Binomial Theorem | |||||||||||
\(\ds \) | \(=\) | \(\ds a^p + 0_F + b^p\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds a^p + b^p\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \phi a + \map \phi b\) |
Multiplication is more straightforward:
\(\ds \map \phi {a b}\) | \(=\) | \(\ds \paren {a b}^p\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds a^p b^p\) | Power of Product of Commutative Elements in Monoid | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \phi a \map \phi b\) |
Thus, $\phi$ is a (field) homomorphism.
It is clear that $\phi$ is not a zero homomorphism, since:
\(\ds \map \phi {1_F}\) | \(=\) | \(\ds 1_F^p\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1_F\) | Power of Identity is Identity | |||||||||||
\(\ds \) | \(\ne\) | \(\ds 0_F\) |
Hence, from Ring Homomorphism from Field is Monomorphism or Zero Homomorphism, it follows that $\phi$ must be a monomorphism.
$\blacksquare$
Also see
- Prime Power of Sum Modulo Prime, where the same technique is used.
Sources
- 1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 3$. Homomorphisms: Theorem $3.3$
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $3$: Field Theory: Definition and Examples of Field Structure: $\S 89 \gamma$