I am considering asking something like this and I am wondering if this would be considered a duplicate question. At this point I haven't really shown sufficient attempts but even if I did I might consider it a duplicate.


As in [1] is entitled: "Constructing an infinite chain of subfields of 'hyper' algebraic numbers?"

we let $S_F$ denote the set of values which satisfy some generalized polynomial whose exponents and coefficients are drawn from $F$. That is, we let $S_F$ denote

$$\bigg \{x \in \mathbb{R}: 0=\sum_{i=1}^n{a_i x^{e_i}}: e_i \in F \text{ distinct}, a_i\in F \text{ non-zero}, n\in \mathbb{N} \bigg \}$$

I want to focus a little on the Algebraic properties of $S_F$. For example,

Main Question

Is it possible to see whether $S_{\mathbb{Q[\sqrt{2}]}}$ is a field? Or is it possible to demonstrate that it is not a field?

My thoughts: Honestly. This borders on a no clue question for me. I am no algebraist. I am not sure what tools I would use to demonstrate that it's a field or not except by going through the field axioms one by one. As was pointed out [1] in the comments: It's hard to even show that $S_F$ is closed under negation. Though $S_F$ does satisfy a few axioms and I can enumerate these.

As a type of meta question: What tools should I be investigating to give me some power over this?

My Motivation: See the previous post on this topic.


Would you rule this as a dupe? Why? Should I delete the "curiosities" from the other post (I mean [1]) if I choose to do this? I would think not but maybe this is logical way to separate them as two distinct questions.

  • $\begingroup$ Also is there a general ruling on these "ask to ask" type meta questions. I guess I am asking about meta: making this a meta meta question. I mean this is a good way of learning the culture of MSE but it seems like it's annoying to evaluate every question as appropriate or inappropriate and we already have the reputation system for this. $\endgroup$ – Mason Dec 30 '18 at 5:17
  • $\begingroup$ Should I evaluate downvotes as "This is a dupe?" That would make the most sense to me. $\endgroup$ – Mason Dec 30 '18 at 5:19
  • $\begingroup$ a downvote represents that the community does not approve of your suggestion. $\endgroup$ – Mohammad Zuhair Khan Dec 30 '18 at 6:56
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    $\begingroup$ On you comment, yes, usually it's not a good idea to ask beforehand for a specific question if it is appropriate. Just go ahead and see what happens. If you want, you can add a comment acknowledging your doubt. $\endgroup$ – quid Dec 30 '18 at 11:51
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    $\begingroup$ I find the \begin{question} and \end{question} amusing. They remind me of my first MSE question, when I didn't know that the type of $\LaTeX$ is something called MathJax. Anyway, @Mason: They're not required. $\endgroup$ – Shaun Dec 30 '18 at 15:50

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