Difference between vector spaces and fields

Question

My doubt:

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

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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 -

1. Addition
2. Multiplication

and of course their inverses (Subtraction and Division).

Thus, $\mathbb{R}$ is a field

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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

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• 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?

Answer 1

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.

Answer 2

A field, say $\Bbb R$, has two inner operations $+:\Bbb R^2\rightarrow\Bbb R: (r,s)\mapsto r+s$ and $\cdot: \Bbb R^2\rightarrow\Bbb R:(r,s)\mapsto r\cdot s$ connected by the law of distributivity, i.e., $r\cdot (s+t) = r\cdot s+r\cdot t$.

A vector space $V$ is defined over a field, say $\Bbb R$, and has one inner operation $+:V^2\rightarrow V$ (vector addition) and one outer operation $\cdot: \Bbb R\times V\rightarrow V$ (scalar multiplication). Both operations are connected, e.g., $r\cdot (v+w) = r\cdot v+r\cdot w$ and $(r+s)\cdot v = r\cdot v+s\cdot v$.

So there are big differences.

For instance, consider the set of complex numbers $\Bbb C = \{a+bi\mid a,b\in\Bbb R\}$, where $i$ is the imaginary unit with $i^2=-1$. First, $\Bbb C$ is a field. Second, $\Bbb C$ is a vector space over $\Bbb R$ with basis $1+0i=1$ and $0+1i= i$.

Answer 3

One usually defines a field as a tuple $(F,0,1,\cdot,+,-,\div)$ where $F$ is a set, $0,1\in F$, $\cdot,+$ encode binary operations on $F$, etc...

One usually defines a vector space as a tuple $(V,k,{\bf 0},+,-,\cdot)$, where $V$ is a set, $k$ is a field, ${\bf 0}\in V$, $\cdot : k\times V\to V$ encodes a compatible action of $k$ on $V$, $+,-$ encode binary operations on $V$, etc...

So formally, if $R$ is the set of real numbers, then the field of real numbers if the object $\mathbb R := (R,0,1,\cdot,+,-,\div)$ where $0,1,\cdot,+,-,\div$ are the usual operations that we define on $R$. Now if we want to talk about the vector space of real numbers we take $(\mathbb R,\mathbb R,0,+,-,\cdot)$ where here we are allowed to take the field operations and reuse them for the vector space because they are the same. However this is not the case if we wish to view the complex numbers as a vector space over $\mathbb R$, for example we would have $$(\mathbb C,\mathbb R,(0,0),+',-',\cdot')$$ where $(0,0)$ is a tuple of real numbers (assuming $\mathbb C$ is defined as pairs of real+imaginary numbers), and $+',-', \cdot'$ are new operations which are compatible between $\mathbb C$ and $\mathbb R$.

In reality though these encodings are ignored and very much abused by mathematicians (even in this post, where even though a field $k=(F,\ldots)$ is a structure, we still allow ourselves to talk about the membership relation $x\in k$ to mean $x\in F$). So when someone says that "The object $X$ is an $A$-structure, and a $B$-structure", what they mean is that the object $X$ can be attributed an $A$-structure or a $B$-structure. A mathematician will say that these are "the same", although they are not formally equal.