Field Homomorphism Preserves Subfields/Corollary

Corollary to Field Homomorphism Preserves Subfields

Let $\struct {F_1, +_1, \circ_1}$ and $\struct {F_2, +_2, \circ_2}$ be fields.

The image of a field homomorphism $\phi: F_1 \to F_2$ is a subfield of $F_2$.

Proof

From Field is Subfield of Itself, $F_1$ is a subfield of $F_1$.

The result then follows directly from Field Homomorphism Preserves Subfields.

$\blacksquare$

Sources

• 1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 3$. Homomorphisms