Structure of a Vector Space through ring homomorphism My question is:
Suppose we have a ring homomorphism from $\mathbb Z$ to any finite field $F$ whose kernel is $p\mathbb Z$ (say).
Now this ring homomorphism induces another ring homomorphism $h:\mathbb Z / p\mathbb Z\to F$.
My question is how this new homomorphism $h$ gives $F$ a structure of a vector space over the field $\mathbb Z/p\mathbb Z$?
 A: Define map $\varphi\colon\mathbb Z/p\mathbb Z\to \operatorname{End}(F)$ with $\bar n\mapsto(x\mapsto h(\bar n)x)$. Verify that $x\mapsto h(\bar n)x$ really is additive so the map is well-defined and that $\varphi$ is ring homomorphism that preserves unit. This defines ring action of $\mathbb Z/p\mathbb Z$ on $F$, and thus $F$ is $\mathbb Z/p\mathbb Z$ vector space.
Alternatively, define multiplication by scalar with $\bar n\cdot x = h(\bar n)x$ and verify vector space axioms.
A: For $c \in \mathbb{Z}/p\mathbb{Z}$ and $a \in F$, define the scalar multiplication to be $c\cdot a = h(c)a.$ You should check that this is a well-defined scalar multiplication.
Any finite field is a finite-dimensional vector space over its prime subfield. Here, you have embedded $\mathbb{Z}/p\mathbb{Z}$ into $F$ as its prime subfield.
A: $\newcommand{\Z}{\mathbb{Z}}$$\renewcommand{\phi}{\varphi}$$\DeclareMathOperator{\End}{End}$Perhaps it is useful to rephrase first the axioms of vector spaces in the following manner.
Let $V$ be an abelian group, and $K$ a field. Let $\End(V)$ be the ring of endomorphisms of $V$.
Then a structure of a $K$-vector space on $V$ is equivalent to a homomorphism of rings with unity $\phi : K \to \End(V)$. In fact, if you have such a structure, and $\lambda v$ denote scalar multiplication of $v \in V$ by $\lambda \in K$, then $\phi(\lambda) = (v \mapsto \lambda v)$ defines such a homomorphism. And conversely if one has such a homomorphism, then defining $\lambda v = (\phi(\lambda)) (v)$ yields a vector space structure.
So in your case $F$ is an abelian group. If $a \in \Z$, the map $v \mapsto a v$ (where $a v$ is the $a$-th multiple of $v \in F$) is an endomorphism of the abelian group $F$, and the map $\Z \mapsto (v \mapsto a v)$ is a homomorphism of rings with unity $\Z \to \End(F)$, which has kernel $p \Z$, and thus yields as you note a homomorphism of rings with unity $\Z/p\Z \to \End(F)$.
