# Category:Field Homomorphisms

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This category contains results about **Field Homomorphisms**.

Definitions specific to this category can be found in Definitions/Field Homomorphisms.

Let $\struct {F, +, \times}$ and $\struct {K, \oplus, \otimes}$ be fields.

Let $\phi: F \to K$ be a mapping such that both $+$ and $\times$ have the morphism property under $\phi$.

That is, $\forall a, b \in F$:

\(\text {(1)}: \quad\) | \(\ds \map \phi {a + b}\) | \(=\) | \(\ds \map \phi a \oplus \map \phi b\) | |||||||||||

\(\text {(2)}: \quad\) | \(\ds \map \phi {a \times b}\) | \(=\) | \(\ds \map \phi a \otimes \map \phi b\) |

Then $\phi: \struct {F, +, \times} \to \struct {K, \oplus, \otimes}$ is a field homomorphism.

## Subcategories

This category has the following 4 subcategories, out of 4 total.

### F

- Field Automorphisms (1 P)
- Field Monomorphisms (4 P)

## Pages in category "Field Homomorphisms"

The following 6 pages are in this category, out of 6 total.