When, exactly, does the degree of an extension of the rationals by two algebraic numbers equal the product of the degrees of the extensions by each number? More precisely, can we give necessary and sufficient conditions for the equation

$$\left[\Bbb Q(\alpha,\beta):\Bbb Q\right]=\left[\Bbb Q(\alpha):\Bbb Q\right]\left[\Bbb Q(\beta):\Bbb Q\right]$$

to hold, where $\alpha$ and $\beta$ are algebraic numbers?

I encountered this condition in an answer at math overflow. I know a little bit: if $\alpha$ and $\beta$ come from different quadratic fields, for example, then the statement is true. If the intersection of $\Bbb Q(\alpha)$ and $\Bbb Q(\beta)$ is non-trivial, then the statement is false. If the two degrees are relatively prime, then the statement is true.

What's the best we can say about this? Have I made any mistakes in my above claims? Thanks in advance.

  • 4
    $\begingroup$ This condition is stated as "$\Bbb Q(\alpha)$ and $\Bbb Q(\beta)$ are linearly disjoint over $\Bbb Q$". $\endgroup$ – Lord Shark the Unknown Dec 26 '17 at 15:36

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