# Addition of two algebraic integer numbers [duplicate]

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.

• @KanwaljitSingh, that question is not about algebraic integers. – lhf Apr 7 '17 at 17:12
• Yep, that answers. Dang I was putting my thoughts into words for this. – fleablood Apr 7 '17 at 17:13
• 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. – fleablood Apr 7 '17 at 17:17
• This question is most likely a duplicate but I couldn't find one with an actual answer. – lhf Apr 7 '17 at 17:26
• Note that this answer contains a proof of this fact, although the question is a bit different. – Lukas Heger Apr 7 '17 at 18:07

• $\alpha$ 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.