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Which of the following statesment are true?

  1. there exists a finite field in which additive group is not cyclic
  2. $F$ is a finite field then there exist a polynomial $p$ over $F$ such that $p(x) \ne0$ for all $x\in F$, where $0$ denotes zeros of $F$
  3. Every finite field is isomorphic to a subfield of the field of complex numbers
trying
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Arib Gullu
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  • What are you thoughts on this? Where did you run into this question? I think it's time you try and dispel the inevitable thought that you are trying to outsource a homework problem here. I warmly recommend that you familiarize yourself with the local guidelines of what is expected from a question. – Jyrki Lahtonen Sep 06 '17 at 14:49
  • Another thing is that you should not ask more than one question per post. I randomly picked one of yours, and found a duplicate. – Jyrki Lahtonen Sep 06 '17 at 14:51
  • @Jyrki How can this question possibly be a duplicate of the linked one? Please don't abuse moderator closing powers like that. It is not your prerogative to "randomly pick" part of the question but censor others. – Bill Dubuque Sep 06 '17 at 15:49
  • @Jyrki Also, it is rather rude to accuse a questioner of "outsourcing homework". For all we know the questioner could be self-studying. This is very poor behavior for a "moderator" (they should exhibit exemplary behavior which, of course, does not include rude or abusive behavior). – Bill Dubuque Sep 06 '17 at 15:54
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    @BillDubuque I view this as an opportunity for the asker to clarify which question they want answered. And since when is a multipart question like this clearly not homework, pray tell me. And this is no more abuse of my power than your late exhibits of abuse of "trusted" user power in dupe closing questions that let your own answer survive. – Jyrki Lahtonen Sep 06 '17 at 16:23
  • @jyrki Alas, we'll have to strongly disagree on these matters (and I have no idea what the second red herring is supposed to mean - it does not appear to be grounded in reality nor does it have anything to do with the matter at hand). Please reopen teh question and let it go through the normal processes. – Bill Dubuque Sep 06 '17 at 16:25
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    @BillDubuque I also need to protect the interests of the answerers, so that they know which question they need to answer. We have had many disappointed users who answered a single part perfectly, but where left out in the voting. – Jyrki Lahtonen Sep 06 '17 at 16:26

1 Answers1

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1. is true: Is there a finite field in which the additive group is not cyclic? Actually every finite field other then $\mathbb{Z}_p$ has non-cyclic additive group.

2. is true: the obvious choice is a constant polynomial $p(x)=1$. But there are also such polynomials of positive degree. If $F=\{a_1,\ldots, a_n\}$ then define $$p(x)=(x-a_1)\cdots(x-a_n)+1$$

3. Is false: $\mathbb{C}$ is a field of characteristic $0$. Every finite field has a positive characteristic. Which means that no finite field embeds as a subfield into $\mathbb{C}$.

freakish
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