Q:

Difference between vector spaces and fields

My doubt:
What is the main difference between a field and a vector space?

In my mind, a field is simply a collection of eveything that follows a certain property. And it also comes with some built-in operations that you can perform on its elements.
For example, the field $\mathbb{R}$ is a collection every number that follows the property of being real.
It comes with two operations -

Addition
Multiplication

and of course their inverses (Subtraction and Division).
Thus, $\mathbb{R}$ is a field

But in my mind, vector spaces also have the exact same definition :-
A collection of elements that has a few built-in operations.
So $\mathbb{R}$ is a vector-space then

Which of the above is true? Is it a field or a vector space?
Also, what is $\mathbb{C}$ then? I read somewhere that is a vector space over $\mathbb{R}$. I also read that it is a field. Which one is correct?

A:

Many objects in mathematics have structure that make them instances of two, and often more, different concepts. The real numbers $\mathbb R$ is an example. As you point out, the reals with the usual structure satisfy the axioms of a field and thus $\mathbb R$ is indeed a field. At the same time, the reals with the usual structure also satisfy the axioms of a vector space (over the reals) and so $\mathbb R$ is a vector space. For that matter, it is also a group, also an abelian group, also a monoid, also a module, and many other things.
$\mathbb C$ is a also a field and it is at the same time a vector space. In fact, it is a vector space both over the reals and over the complex numbers. As a vector space over the reals it has dimension $2$ while over the complex numbers it has dimension $1$.
So, your observation is correct about $\mathbb R$. There should be no problem at all with one given structure being an example of many different abstract concepts. This happens often and it is very fruitful to realise this.

