proof that Q(sqrt(2)) is a subfield of R
Q= rational numers, R= real numbers.
The Attempt at a Solution
Clearly (sqrt(2)) is a subgroup of R. Then a[sqrt2]. b[sqrt2] are elements of Q[sqrt2] if a and b are eleemnts of Q. Therefore Q[sqrt2] contains at least two elements.
2. a[sqrtb]-b[sqrt2] is an element of Q[sqrt2] since a-b is an element of Q since Q is closed under subtraction.
From there I have to prove that a[sqrt2]*[b[sqret2]^-1] are elements of Q[sqrt2] but i don't know how to do this. Any help would be appreciated.