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Let F be a field and $E_{1}$ $E_{2}$, two extensions of F, does exist a criteria for proving that they are isomorphic as fields? I know that in particular cases like for example with F = $\mathbb{Q}$, $E_{1}$ = $\mathbb{Q}$($\sqrt 2 $) and $E_{2}$ = $\mathbb{Q}$($\sqrt 7 $), $E_{1}$ and $E_{2}$ aren't isomorphic because $\sqrt 2$ $\notin$ $\mathbb{Q}$($\sqrt 7 $), how this can be generalized?

Turquoise Tilt
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    This is hard in general. Though you have special cases like https://math.stackexchange.com/questions/246648/isomorphism-of-two-field-extensions?rq=1 – wormram Oct 23 '20 at 18:55

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