- #1
nonequilibrium
- 1,439
- 2
By F[X] I mean the polynomials with coefficients in field F. By F(X) I mean the rational polynomials.
I have a feeling that [itex]\boxed{ \mathbb Q( \sqrt 2 ) \cong \frac{\mathbb Q[X]}{(X^2-2)}} [/itex]. (if not readable: the RHS is with [X])
Is this true? If so, how can I prove it? I suppose it would suffice I could show that the RHS is the smallest field extension of the rationals that contains sqrt(2) (as the LHS is obviously just that).
Also, is there maybe even a more general result behind this?
I have a feeling that [itex]\boxed{ \mathbb Q( \sqrt 2 ) \cong \frac{\mathbb Q[X]}{(X^2-2)}} [/itex]. (if not readable: the RHS is with [X])
Is this true? If so, how can I prove it? I suppose it would suffice I could show that the RHS is the smallest field extension of the rationals that contains sqrt(2) (as the LHS is obviously just that).
Also, is there maybe even a more general result behind this?