Linear maps and field maps between finite fields

Statement

Fix a prime $p$ and positive integers $a,b$ . Choosing bases of the finite fields over $\mathbb F_p$ identifies their linear maps with rectangular matrices:

\operatorname{Hom}_{\mathbb F_p\text{-Vect}}(\mathbb F_{p^a},\mathbb F_{p^b}) \cong M_{b\times a}(\mathbb F_p).

In the category of fields, maps preserve addition, multiplication and $1$ : the same pair instead has $a$ homomorphisms if $a\mid b$ , and none otherwise. The two comparisons use different morphism axioms on the same finite carriers.

Linear endomorphisms and automorphisms

For $a=b$ , the matrix identification respects addition and composition, so the linear endomorphism ring is $M_a(\mathbb F_p)$ , with $p^{a^2}$ elements. Its units are $\mathrm{GL}_a(\mathbb F_p)$ , whose cardinality is

\left|\mathrm{GL}_a(\mathbb F_p)\right| =\prod_{j=0}^{a-1}(p^a-p^j).

Indeed, the first column can be any nonzero vector, and each subsequent column must avoid the span of its predecessors. For a $b\times a$ matrix there are $ab$ independent entries, giving $p^{ab}$ linear maps. The matrix identification depends on the chosen bases.

Concrete comparisons

Object	Scalar field	Linear endomorphisms	Linear automorphisms	Field endomorphisms = field automorphisms
$\mathbb F_4$	$\mathbb F_2$	$M_2(\mathbb F_2)$ , 16 maps	$\mathrm{GL}_2(\mathbb F_2)$ , 6 maps	2 Frobenius powers
$\mathbb F_8$	$\mathbb F_2$	$M_3(\mathbb F_2)$ , 512 maps	$\mathrm{GL}_3(\mathbb F_2)$ , 168 maps	3 Frobenius powers
$\mathbb F_9$	$\mathbb F_3$	$M_2(\mathbb F_3)$ , 81 maps	$\mathrm{GL}_2(\mathbb F_3)$ , 48 maps	2 Frobenius powers

For example there are $2^{3\cdot2}=64$ linear maps from $\mathbb F_4$ to $\mathbb F_8$ , but no field homomorphisms, since $2\nmid3$ . The zero linear map is one of those 64 maps and never a field homomorphism.

Why the field-map count is different

A field homomorphism is injective and fixes the prime field. Its image would make $\mathbb F_{p^b}$ an extension of $\mathbb F_{p^a}$ , forcing $a\mid b$ by the tower formula. When $a\mid b$ , the target contains a unique subfield of size $p^a$ , and the isomorphisms onto it differ by the $a$ powers of Frobenius $x\mapsto x^p$ . Thus every endomorphism of a finite field is an automorphism, although most linear endomorphisms are not even injective.

Fixed scalar categories

The catalogue uses distinct categories $\mathbb F_2\text{-Vect}$ and $\mathbb F_3\text{-Vect}$ . A symbolic $\mathbb F_p\text{-Vect}$ comparison requires the same fixed prime $p$ and the specified prime-field scalar actions at both endpoints. A shared letter standing for an unspecified prime does not create a linear Hom-set across different characteristics.

References

J. S. Milne, Fields and Galois Theory, Chapter 4, Proposition 4.20, Corollary 4.21 and Proposition 4.23, pp. 53–54.