Field Homomorphism is either Trivial or Injection
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Theorem
Let $\struct {E, +_E, \times_E}$ and $\struct {F, +_F, \times_F}$ be fields.
Let $\phi: E \to F$ be a (field) homomorphism.
Then $\phi$ is either an injection or the trivial homomorphism.
Proof 1
This is an instance of Ring Homomorphism from Field is Monomorphism or Zero Homomorphism.
$\blacksquare$
Proof 2
Let $\phi: E \to F$ be a field homomorphism.
Suppose $\phi$ is not an injection.
So there must exist $a, b \in F: \map \phi a = \map \phi b$.
Let $k = a +_E \paren {-b}$.
Then:
\(\ds \map \phi k\) | \(=\) | \(\ds \map \phi {a +_E \paren {-b} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \phi a +_F \map \phi {-b}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \phi a +_F \paren {-\map \phi b}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 0_F\) | as $\map \phi a = \map \phi b$ |
As $a \ne b$ then $k \ne 0_E$ and so has a product inverse $\exists k^{-1} \in E$.
So for any $x \in E$ we can write $x = k \circ \paren {k^{-1} \circ x}$ and so:
\(\ds \map \phi x\) | \(=\) | \(\ds \map \phi {k \times_E \paren {k^{-1} \times_E x} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \phi k \times_F \map \phi {k^{-1} \times_E x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 0_F \times_F \map \phi {k^{-1} \times_E x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 0_F\) |
So if $\phi$ is not an injection, it is the trivial homomorphism.
$\blacksquare$
Sources
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $3$: Field Theory: Definition and Examples of Field Structure: $\S 87 \eta$