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?