Question: In learning about Vector Spaces over a Field, I have come accross the notation: ($\mathbb{K}$, $\cdot$, +) is a Field. My Question is, what in this context is $\mathbb{K}$?

The question seems very simple, and part of me wants to just guess that $\mathbb{K}$ is just a set and the set combined with the two operations (and axioms) forms the field, but google doesn't really seem to know what I am asking.

Thanks!

Answer: 
$\Bbb K$ is a set, $\cdot$ and $+$ are binary operations on this set, i.e., maps $\Bbb K\times \Bbb K\to \Bbb K$, $(x,y)\mapsto x\cdot y$ (or $\mapsto x+y$, respectively). Additional properties (specific laws for the addition and multiplication, special roles of special elements of $\Bbb K$) are then postulated in form of axioms.

For example, $\Bbb K$ might be the set of all rational numbers, and $+$ and $\cdot$ might be the usual addition and multiplication for rational numbers (which, as desired, result in a rational number again).
There appears to be some confusion surrounding the notation itself, so first, I should clarify what is happening. The symbol $(\mathbb{K},0,1,+,\cdot)$ denotes a mathematical structure, where

* $\mathbb{K}$ is some arbitrary class of objects (in $\mathrm{ZFC},$ a set).
* $0\in\mathbb{K},\;1\in\mathbb{K},\;0\neq1$
* $+$ is a function $\mathbb{K}^2\to\mathbb{K}$ with inputs $(x,y)$ and outputs $z.$ Typically, we denote this by $z=x+y.$
* $\cdot$ is a function $\mathbb{K}^2\to\mathbb{K}$ with inputs $(u,v)$ and outputs $w.$ Typically, we denote this by $w=u\cdot{v}.$

If this structure satisfies the following axioms:

* $x+y=y+x$ and $x\cdot{y}=y\cdot{x}$
* $x+(y+z)=(x+y)+z$ and $x\cdot(y\cdot{z})=(x\cdot{y})\cdot{z}$
* $0+x=x+0=x$
* $1\cdot{x}=x\cdot1=x$
* For every $x,$ there exists a corresponding $-x$ such that $x+(-x)=(-x)+x=0$
* For every $x\neq0,$ there exists a corresponding $x^{-1}$ such that $x\cdot{x}^{-1}=x^{-1}\cdot{x}=1$
* $x\cdot(y+z)=x\cdot{y}+x\cdot{z}$

then the mathematical structure is called a **field.** However, in an abuse of notation, we tend to simply denote the field itself with the symbol $\mathbb{K},$ instead of the full $(\mathbb{K},0,1,+,\cdot).$

For example, $(\mathbb{R},0,1,+,\cdot)$ denotes the field of real numbers. However, in practice, almost everyone just writes $\mathbb{R}$ to denote the field of real numbers. This is what needs to be understood. The set $\mathbb{R}$ itself is not relevant. You can replace the elements of $\mathbb{R}$ bijectively with other mathematical objects, but as long as the field axioms are still satisfied, after the replacement and all the theorems are appropriately modified, you still have the same field, so what the set $\mathbb{R}$ itself is does not matter. You only need a set that is *representative* of what the field looks like.

Now, in the definition of a vector space, which field you choose as the scalar field is arbitrary, and it just depends on what you are doing. You can define a vector space over $\mathbb{Z}\_2,$ over $\mathbb{Q},$ over $\mathbb{R},$ or over $\mathbb{C},$ or over any other field. In most applications though, the field chosen is $\mathbb{R}$ or $\mathbb{C}.$