I am looking for a hint on the following problem: Find fields $K_1$ and $K_2$ with $$ \mathbb{Q}{(\sqrt{3})}\otimes_\mathbb{Q}\mathbb{Q}(\sqrt[4]{3})\cong K_1\times K_2 $$ My thoughts were to try and use the universal property of tensor products, by maybe defining map $(a,b)\mapsto ab$ from $\mathbb{Q}{(\sqrt{3})}\times\mathbb{Q}(\sqrt[4]{3})\to \mathbb{Q}(\sqrt[4]{3})$ which would induce a linear map on the tensor product. The problem here is this is not (at least obviously to me) a product of fields, nor an isomorphism.
Any hints would be appreciated.
Well, you know that $\mathbb{Q}[\sqrt{3}] \simeq \mathbb{Q}[x]/(x^2-3)$. Also, for fields (and for more general rings too I think, but maybe there are some conditions on the rings) $K[x] \otimes L = L[x] $. Hope that helps! Edited for more details: I believe this is correct: Let $K=\mathbb{Q}[\sqrt{3}]$. Then we want $K[x]/(x^4-3) = K[x]/(x^2-\sqrt{3}) \oplus K[x]/(x^2+\sqrt{3})=\mathbb{Q}(3^{\frac{1}{4}}) \oplus \mathbb{Q}(3^{\frac{1}{4}})$.