Addition of two algebraic integer numbers

Question

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 07 '17 at 17:12
Yep, that answers. Dang I was putting my thoughts into words for this. — fleablood, Apr 07 '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 07 '17 at 17:17
This question is most likely a duplicate but I couldn't find one with an actual answer. — lhf, Apr 07 '17 at 17:26
Note that this answer contains a proof of this fact, although the question is a bit different. — Lukas Heger, Apr 07 '17 at 18:07
@MatheiBoulomenos But it openly states that the aim is to prove the set of algebraic integers is a ring. — egreg, Apr 08 '17 at 17:07

score 3 · Answer 1 · answered Apr 07 '17 at 17:21

3

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.

answered Apr 07 '17 at 17:21

lhf

See also https://en.wikipedia.org/wiki/Resultant#Number_theory and https://en.wikipedia.org/wiki/Algebraic_integer#Facts for a different take. – lhf Apr 07 '17 at 17:23

score 0 · Answer 2 · answered Apr 07 '17 at 17:13

0

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

answered Apr 07 '17 at 17:13

Flasgod

2

The question is about algebraic integers and whether they form a ring. – lhf Apr 07 '17 at 17:15
oh wow, absolutely mis-read the question. I'm so sorry. A quick search should give Helia.alipanah a pretty good insight anyway.
https://en.wikipedia.org/wiki/Algebraic_integer
– Flasgod Apr 07 '17 at 22:02

2 Answers2