Compute all intermediate fields of \[ L=\QQ(\sqrt2,\sqrt3)/\QQ. \] Show that \[ L=\QQ(\sqrt2+\sqrt3) \] and find the minimal polynomial of \(\sqrt2+\sqrt3\) over \(\QQ\).
Solution
The extension is biquadratic: \[ [L:\QQ]=4, \qquad \operatorname{Gal}(L/\QQ)\cong C_2\times C_2. \] The three subgroups of order \(2\) correspond to the three quadratic intermediate fields \[ \QQ(\sqrt2), \qquad \QQ(\sqrt3), \qquad \QQ(\sqrt6). \] Together with \(\QQ\) and \(L\), these are all intermediate fields.
Set \[ \alpha=\sqrt2+\sqrt3. \] Since \[ (\sqrt3+\sqrt2)(\sqrt3-\sqrt2)=1, \] we have \[ \alpha^{-1}=\sqrt3-\sqrt2. \] Therefore \[ \sqrt3=\frac{\alpha+\alpha^{-1}}2, \qquad \sqrt2=\frac{\alpha-\alpha^{-1}}2, \] so \[ \QQ(\alpha)=\QQ(\sqrt2,\sqrt3)=L. \]
Now \[ \alpha^2=5+2\sqrt6, \] so \[ (\alpha^2-5)^2=24. \] Hence \[ \alpha^4-10\alpha^2+1=0. \] Thus \(\alpha\) is a root of \[ f(x)=x^4-10x^2+1. \] Because \(\QQ(\alpha)=L\) has degree \(4\) over \(\QQ\), this degree-\(4\) polynomial is the minimal polynomial.