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
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Expert Answer

coffentw
Answered 2020-10-24 Author has 103 answers

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 KL is a subfield of L if K is a field, with the same addition and multiplcation operations as in L.
Examples:
1)Q,R are subfields of R,C respectively.
2)Q(p),pprime, is a subfield of RR.
Let L be a field.
Let {Ki,iI} be any collection of subfields Ki of L.
Claim: K=iIKi is a field.
Proof: KL,asKiLiI.
x,yKx+y,xy,0,1,xyKi,iI
x+y,xy,0,1,xyiIKi
x+y,xy,0,1,xyK
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, xK,x0,xKi,iI,x0
rArr x1Ki,iI (as Ki are all fields)
rArr x1iIKi=K
Thus, K is a field, in fact, K is a subfield of L

Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question