# Tensor product of fields and ramification theory

"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.)"
• "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. May 26, 2013 at 19:29