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

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

2020-10-24
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)$$\displaystyle\mathbb{Q}{\left(\sqrt{{p}}\right)},{p}{p}{r}{i}{m}{e},$$ is a subfield of RRZSK.
Let L be a field.
Let $$\displaystyle{\left\lbrace{K}_{{i}},{i}\in{I}\right\rbrace}$$ be any collection of subfields $$\displaystyle{K}_{{i}}$$ of L.
Claim: $$\displaystyle{K}=\bigcap_{{{i}\in{I}}}{K}_{{i}}$$ is a field.
Proof: $$\displaystyle{K}\subset{L},{a}{s}{K}_{{i}}\subset{L}\forall{i}\in{I}.$$
$$\displaystyle{x},{y}\in{K}\Rightarrow{x}+{y},{x}-{y},{0},{1},{x}{y}\in{K}_{{i}},\forall{i}\in{I}$$
$$\displaystyle\Rightarrow{x}+{y},{x}-{y},{0},{1},{x}{y}\in\bigcap_{{{i}\in{I}}}{K}_{{i}}$$
$$\displaystyle\Rightarrow{x}+{y},{x}-{y},{0},{1},{x}{y}\in{K}$$
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, $$\displaystyle{x}\in{K},{x}\ne{0},\Rightarrow{x}\in{K}_{{i}},\forall{i}\in{I},{x}\ne{0}$$
rArr $$\displaystyle{x}^{{-{{1}}}}\in{K}_{{i}},\forall{i}\in{I}$$ (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

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