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Suppose $U,V,W$ are the cyclotomic fields $\mathbb{Q}(\zeta_{m_3})$, $\mathbb{Q}_(\zeta_{m_2})$, and $\mathbb{Q}(\zeta_{m_1})$ respectively, where $m_1 \mid m_2 \mid m_3$ so that $U/V/W$ is a tower of fields.

It is true that if $\{u_i\}$ is a $V$-basis for $U$ and $\{v_i\}$ is a $W$-basis for $V$ then $\{u_i\cdot v_j\}$ is a $W$-basis for $U$. My question is about how I should view this fact in a more general setting.

I found a statement about vector spaces that seems pretty close: if $A$ and $B$ are vector spaces over a field $F$ with ($F$-?)bases $\{a_i\}$ and $\{b_i\}$ respectively, then $\{a_i\otimes b_j\}$ is an ($F$-?)basis for $A\otimes B$.

My cyclotomic example almost fits into that setting: $U$ is a f.d. vector space over $V$ with basis $\{u_i\}$ and $V$ is a f.d. vector space over $W$ with basis $\{v_i\}$. We have $U\otimes_V V\cong U$ (since $m_2=gcd(m_2,m_3)$ and $m_3=lcm(m_2,m_3)$) under the isomorphism $(a\otimes b) \mapsto ab$, and, as I mentioned above, $\{u_i \otimes v_j\}=\{u_i\cdot v_j\}$ is a basis for $U\otimes_V V$.

The problem is that the statement I found about vector spaces seems to apply when the bases for the vector spaces are $F$-bases, whereas I am using relative bases. Is this fact about cyclotomic fields true in some more general setting?

Bonus question: What if $U,V,W$ are cyclotomic rings, i.e., $\mathbb{Z}[\zeta_m]$? The fact is still true: the tensor of relative bases is a $W$-basis for $U$, but I can't use the vector space setting since I don't have a field. This seems like more evidence that this fact can be put in a more general setting.

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