Write a theory $\gamma$ so that its models $M$ are exactly the vector spaces over $\mathbb{F}$

Question

Write a theory $\gamma$ so that its models $M$ are exactly the vector spaces over $\mathbb{F}$, in the language $L=\{0, +, (\mu_a) a\in \mathbb{F}\}$, where $u_a$ represents binary scalar addition by an element $a$. And $+$ is vector addition.

So I have the axioms of a vector space, and I need to convert symbolize them I believe.

1. Commutativity $\forall x\forall y(x+y=y+x)$

2. Associativity of addition: $\forall x\forall y\forall z((x+y)+z=x+(y+z))$

3. Additive Identity: $\forall x (x+0=x)$

4. Additive inverse: $\forall x \exists y (x+y=0)$

5. Associativity of scalar multiplication: $\forall x (u_a(u_b (x))=u_{c}(x))$

6. Distributivity of scalar: $\forall x ((u_a+u_b)(x)=u_a(x)+u_b(x))$

7. Distributivity of addition: $\forall x \forall y (u_a(x+y)=u_a(x)+u_a(y))$

8. Scalar identity: $\forall x ( u_1(x)=x)$

My questions are, I'm unsure of the axioms involving scalar multiplication. I assume I need a separate sentence for every element of the field?

Answer 1

Exactly, 5. is an axiom schema : for each $a,b \in \mathbb{F}$, you have the axiom $$\forall x \ u_a(u_b(x)) = u_{ab}(x)$$

Notice that 6. and 7. are axiom schemata as well. Btw you need to rewrite 6. :
$$\{\forall x \ u_{a+b}(x) = u_a(x) + u_b(x) \ \big| \ a,b\in\mathbb{F}\}$$.