Why are fields denoted by $\mathbb{K}$?

Question

In a lot of papers and books, $\mathbb{K}$ means $\mathbb{R}$ or $\mathbb{C}$.

I know that $\mathbb{R}$ comes from the word "real", and $\mathbb{C}$ from the word "complex". But what about $\mathbb{K}$?

And why is it better to state theorems over $\mathbb{K}$ and not just over $\mathbb{C}$?

Answer 1

The letter $\mathbb{K}$ comes from the German word Körper (body). In Portuguese, for example, a similar word is used, corpo, which also stands for body. People rather state results for $\mathbb{K}$ just because it is more general.

Answer 2

The name used in German for “field” is “Körper”.

Answer 3

Regarding the question as to whether it's better to state a theorem over a general field $\Bbb K$ than over $\Bbb R$ or $\Bbb C$: it really depends on the result, and how general it is (that is, for which fields it holds). For example, the following theorem is true for any field, so it's well expressed in terms of the generic $\Bbb K$:

Theorem. Let ${\Bbb K}[x]$ be a polynomial ring over a field ${\Bbb K}$. Suppose $f\in{\Bbb K}[x]$ has a root $a\in \Bbb K$. Then $(x-a)$ divides $f$.

However, the following theorem expresses the fact that $\Bbb C$ is algebraically closed; it is not true if $\Bbb C$ is replaced by $\Bbb R$.

Theorem. Let $f\in{\Bbb C}[x]$ have degree $n>0$ and leading coefficient $1$. Then $f$ factors as $f(x) =(x-a_1)\cdots(x-a_n)$ and this factorization is unique up to permutation of the factors.