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Is the addition of two algebraic integer numbers also algebraic?

I, guess it is, but i can't prove it. I wonder if multiplication of them is also algebraic.

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    $\begingroup$ @KanwaljitSingh, that question is not about algebraic integers. $\endgroup$ – lhf Apr 7 '17 at 17:12
  • $\begingroup$ Yep, that answers. Dang I was putting my thoughts into words for this. $\endgroup$ – fleablood Apr 7 '17 at 17:13
  • $\begingroup$ Hmm, I think it be straightforward (famous last words) that if the leading coefficient of P and Q are one the resultant polynomial will have coefficient 1... I hope. $\endgroup$ – fleablood Apr 7 '17 at 17:17
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    $\begingroup$ This question is most likely a duplicate but I couldn't find one with an actual answer. $\endgroup$ – lhf Apr 7 '17 at 17:26
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    $\begingroup$ Note that this answer contains a proof of this fact, although the question is a bit different. $\endgroup$ – Lukas Heger Apr 7 '17 at 18:07
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Hint:

  • $\alpha$ is an algebraic integer iff $\mathbf Z[\alpha]$ is a finitely generated $\mathbf Z$-module.

  • If $\alpha$ and $\beta$ are an algebraic integers, then $\mathbf Z[\alpha,\beta]$ is a finitely generated $\mathbf Z$-module.

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I mean, there's a Field of algebraic numbers, so yes.

quick look here should provide you with an insight https://en.wikipedia.org/wiki/Algebraic_number#The_field_of_algebraic_numbers

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    $\begingroup$ The question is about algebraic integers and whether they form a ring. $\endgroup$ – lhf Apr 7 '17 at 17:15
  • $\begingroup$ oh wow, absolutely mis-read the question. I'm so sorry. A quick search should give Helia.alipanah a pretty good insight anyway. en.wikipedia.org/wiki/Algebraic_integer $\endgroup$ – jorgeegroj Apr 7 '17 at 22:02

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