Let $X$ be a variable, $ς$ a cubic root of the unit and $\sqrt[3]{2}$ the real cubic root of $2$. Let $k: = \mathbb{Q} (ς)$ and $K: = k [\sqrt[3]{2}]$. Show that $K\cong k [X] / <X^3-2>$ and $K\otimes_k K\cong K\times K\times K$.
I do not understand what $\mathbb{Q} (ς)$ and $k [\sqrt[3]{2}]$ mean, Could someone explain me please? Would be the polynomials with coefficients in $\mathbb{Q}$ such that its root is $ς$? To show that $K\cong k [X] / <X^3-2>$ We could take an homomorphism $\varphi : k[X]\rightarrow K$ and prove that its nucleus is $<X^3-2>$? I had thought that the function $\varphi$ is the one that takes a polynomial and evaluates it in a number but I do not know which one, could take an arbitrary one? And to test the second part I thought of using the universal property of the tensor product but I do not know which function to define in the Cartesian product, what do you think?