Assume Q[sqrt(2)] is a subfield of the Real field R, and prove that it is the smallest subfield of R that contains sqrt(2)
Q[sqrt(2)]= a + b(sqrt(2))
The way is usually defined is to be the smallest subfield containing and . Look Here.
So I will assume you are defining and you want to show it is the smallest such field. Let be a field containing and . We need to show . Let then but then because it is closed under sums and products, thus, .
Because a field has to be closed under sums and products (that is basically the definition of "field"). Meaning if then .This is JCIR you answered my question on fields and subfields, can you tell my why is it closed under sums and products ( i need to prove that too)
Thanks alot for your help.