 # Write down a definition of a subfield. Prove that the intersection of a set of subfields of a field F is again a field Mylo O'Moore 2020-10-23 Answered
Write down a definition of a subfield. Prove that the intersection of a set of subfields of a field F is again a field
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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 $K\subset L$ is a subfield of L if K is a field, with the same addition and multiplcation operations as in L.
Examples:
1)$\mathbb{Q},\mathbb{R}$ are subfields of $\mathbb{R},\mathbb{C}$ respectively.
2)$\mathbb{Q}\left(\sqrt{p}\right),pprime,$ is a subfield of RR.
Let L be a field.
Let $\left\{{K}_{i},i\in I\right\}$ be any collection of subfields ${K}_{i}$ of L.
Claim: $K=\bigcap _{i\in I}{K}_{i}$ is a field.
Proof: $K\subset L,as{K}_{i}\subset L\mathrm{\forall }i\in I.$
$x,y\in K⇒x+y,x-y,0,1,xy\in {K}_{i},\mathrm{\forall }i\in I$
$⇒x+y,x-y,0,1,xy\in \bigcap _{i\in I}{K}_{i}$
$⇒x+y,x-y,0,1,xy\in K$
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, $x\in K,x\ne 0,⇒x\in {K}_{i},\mathrm{\forall }i\in I,x\ne 0$
rArr ${x}^{-1}\in {K}_{i},\mathrm{\forall }i\in I$ (as ${K}_{i}$ are all fields)
rArr ${x}^{-1}\in \bigcap _{i\in I}{K}_{i}=K$
Thus, K is a field, in fact, K is a subfield of L