Write down a definition of a subfield. Prove that the intersection of a set of subfields of a field F is again a field

To define the concept of a subfield of a field and prove the stated property regarding subfields of a field.
A subfield of a field L is a subset K of L , which is also a field, with field structure inherited from L.
Let L be a field.
Definition:
A subset ${\displaystyle K\subset L}$ is a subfield of L if K is a field, with the same addition and multiplcation operations as in L.
Examples:
1)${\displaystyle \mathbb{Q},\mathbb{R}}$ are subfields of ${\displaystyle \mathbb{R},\mathbb{C}}$ respectively.
2)${\displaystyle \mathbb{Q}\left(\sqrt{p}\right),pprime,}$ is a subfield of RR.
Let L be a field.
Let ${\displaystyle \{K_i,i\in I\}}$ be any collection of subfields ${\displaystyle K_i}$ of L.
Claim: ${\displaystyle K=\bigcap _{i\in I}{K}_i}$ is a field.
Proof: ${\displaystyle K\subset L,as{K}_i\subset L\mathrm{\forall }i\in I.}$
${\displaystyle x,y\in K\Rightarrow x+y,x-y,0,1,xy\in K_i,\mathrm{\forall }i\in I}$
${\displaystyle \Rightarrow x+y,x-y,0,1,xy\in \bigcap _{i\in I}{K}_i}$
${\displaystyle \Rightarrow x+y,x-y,0,1,xy\in K}$
So, K is a commutive ring with 1
ANSWER: proved that the intersection of any collection of subfields of a field L is indeed a field, (in fact , a subfield of L)
Also, ${\displaystyle x\in K,x\ne 0,\Rightarrow x\in K_i,\mathrm{\forall }i\in I,x\ne 0}$
rArr ${\displaystyle x^{-1}\in K_i,\mathrm{\forall }i\in I}$ (as ${\displaystyle K_i}$ are all fields)
rArr ${\displaystyle x^{-1}\in \bigcap _{i\in I}{K}_i=K}$
Thus, K is a field, in fact, K is a subfield of L