I am considering Exercise 13.2.6 from Dummit & Foote's Abstract Algebra:

Prove directly from the definitions that the field $F(\alpha_1, \alpha_2, \dots, \alpha_n)$ is the composite of the fields $F(\alpha_1,), F(\alpha_2), \dots, F(\alpha_n)$.

We want to prove from definitions that $F(\alpha_1, \alpha_2, \dots, \alpha_n) = F(\alpha_1,), F(\alpha_2), \dots, F(\alpha_n)$

I want to proceed by using minimality of the generated field and composite field. By definition, $F(\alpha_1, \dots, \alpha_n)$ is the smallest field which contains $F$ as well as $\alpha_1, \dots, \alpha_n$ and therefore $F(\alpha_1,\dots,\alpha_n) \subseteq F(\alpha_1)\dots F(\alpha_n)$. Similarly, the composite $F(\alpha_1)\dots F(\alpha_n)$ is the smallest field containing each of $F$ and the $\alpha_i$'s so we have the reverse inclusion $F(\alpha_1)\dots F(\alpha_n) \subseteq F(\alpha_1\dots \alpha_n)$ which completes the proof.

I am unsure about this proof because it seems to be rather obvious and straightforward. Sometimes when I've felt this way, it was because I was assuming some important step that required proof. I want to get the community's thoughts: Is there a different way of approaching the problem? Is this one complete?


It feels like you can be more precise, but your proof is essentially correct. It partly depends on the exact definition used for the two fields - "minimality" is a vague term. I prefer to define them each as the intersections of fields satisfying the conditions.

$F(\alpha_1,\cdots,\alpha_n)$ contains $F$ and $\alpha_i$ or $i=1,2,3,\dots,n$, so $F(\alpha_i)\subseteq F(\alpha_1,\cdots,\alpha_n)$, which, by the definition of composite, means $F(\alpha_1)\cdots F(\alpha_n)\subseteq F(\alpha_1,\cdots,\alpha_n)$.

On the other hand, $F(\alpha_1)\cdots F(\alpha_n)$ contains $F(\alpha_i)$, by the definition of composite, and hence contains $F$ and $\alpha_i$ for each $i=1,2,\dots,n$, which, by definition, means $F(\alpha_1,\cdots,\alpha_n)\subseteq F(\alpha_1)\cdots F(\alpha_n)$ by the definition of $F(\alpha_1,\cdots,\alpha_n)$.

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