Example of a union of subfields that is not a field

Question

I'm looking for a field $F$ such that for subfields $K,E$ of $F$ the structure $K\cup E$ is not a field.

Clearly both $K$ and $E$ must be proper subfields, and while looking for the answer I've come across the primitive element theorem, which might have an answer in general to the question of a union of subfields being a field, but I don't know enough mathematics to understand that theorem - currently I'm just trying to find an example.

An explanation about the fundamental reasons why the general principle (that a union of proper subfields is not a field) holds would also be much appreciated!

Answer 1

Examples abound with finite fields. Let $\overline{\mathbb{F}}_{2}$ be an algebraic closure of $\mathbb{F}_{2}$. Consider the subfields $\mathbb{F}_{4}$ and $\mathbb{F}_{8}$. These fields only have $\mathbb{F}_{2}$ in common, so the order of their union is $4 + 8 - 2 = 10$. There is no finite field of order $10$.

Answer 2

Consider $\Bbb Q(\sqrt 2)\cup\Bbb Q(\sqrt 3)\subseteq \Bbb Q(\sqrt 2,\sqrt 3).$ The union does not contain $\sqrt 2+\sqrt 3.$

Answer 3

For a field $F$, the rational function fields $F(x),F(y)\subseteq F(x,y,z)$,

but $F(x)\cup F(y)$ does not contain $xy$.

Answer 4

Take $K:=\Bbb C$, $E=\Bbb R$ and $F:=\{a+ib,a,\in \Bbb Q\}$. Then $e\in E$, $i\in F$ but $e+i\notin E\cup F$.

Actually, we need to have $E\subset F$ or $F\subset E$ , otherwise take $e\in E\setminus F$ and $f\in F\setminus E$; then $e+f\notin E\cup F$.

Answer 5

Plenty of examples already, so I'll just add something in the line of intuition. Any field is an (abelian) group under addition. For arbitrary subgroups $H, K \subseteq G$ of a group $G$, $H \cup K$ is not a group unless $H \subseteq K$ or $K \subseteq H$ (I leave this as an exercise). Hence, the for the union of two fields to be a field, one must contain the other. This statement is also sufficient trivially.