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In the wikipedia page: http://en.wikipedia.org/wiki/Tensor_product_of_fields They write:

"For example if one adjoins √2 to the rational field ℚ to get K, and √3 to get L, it is true that the field M obtained as K.L inside the complex numbers ℂ is (up to isomorphism) $K\otimes_ℚ L$ as a vector space over ℚ. (This type of result can be verified, in general, by using the ramification theory of algebraic number theory.)"

I'm curious about the statement in the brackets. Do they simply mean that one can obtain that the fields are linearily disjoint because, say, the extensions ramify at different primes? Or mabye they mean that I have deeper knowlage of the tensor product by the ramification data?(Can I determine its prime spectrum for instance?)

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    $\begingroup$ "Do they simply mean that one can obtain that the fields are linearly disjoint because, say, the extensions ramify at different primes?" I assume so, yes. In general it is not possible to tell whether two number fields are linearly disjoint over $\mathbb{Q}$ just by looking at ramification data. $\endgroup$ May 26, 2013 at 19:29

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