Show that the sum of 2 algebraic integers is an algebraic integer using resultants

Question

I've started working on algebraic numbers very recently for a memoir, that is I didn't study them in class.

I need them, and particularly algebraic integers, to prove a couple propositions which aren't the topic here. The thing is, I use the fact that a sum of algebraic integers is again an algebraic integer. I thought this wouldn't take too long to prove, however I can't finish my argument.. so any help would be welcome!

Here is what I've done so far:

I considered algebraic integers as roots of unitary polynomials of the ring $\mathbb{Z}[x]$. Let $a,b$ be algebraic integers cancelled by unitary $P,Q \in \mathbb{Z}$, respectively. I've seen several prooves using Field extension or Galois Theory (for example on mathstack Sums and products of algebraic numbers and How to prove that the sum and product of two algebraic numbers is algebraic?), but I haven't studied those, and I believe it can be proved using only resultants, so this is what I'm trying to to.

If $P(x) = \sum\limits_{i=0}^pa_ix^i$ and $Q(x) = \sum\limits_{i=0}^qb_ix^i$, then for $z= a + b$, $Q(z - x) = \sum\limits_{i=0}^qa_i(z-x)^i$.

This way $res_x(P(x),Q(z-x)) = (-1)^{pq}res_x(Q(z-x), R(z-x))$ (because the leading coefficient of $Q$ is 1) where $R$ is such that $P = AQ + R$ and $deg(R) < q$.

Then it seems that if we show this resultant is zero, it means $z$ is an algebraic integer, but why is that true ? And how can we express this resultant to get to this conclusion ?

Thanks in advance for your help

Answer 1

That's really a standard property of Resultants. Let $P$ and $Q$ be two non-zero polynomials with coefficients in a field $F$ . Then $P$ and $Q$ have a common root in some extension of $F$ if and only if $res(P, Q) = 0$. This follows from the product formula for $res(P,Q)$ involving the roots of $P$ and $Q$.

$P(x)$ and $Q(z-x)$ have the common root $a$, so that $res(P,Q)=0$. Then since $a+b$ is a root of $res(P(x),Q(z-x))$ as a polynomial in $z$, $a+b$ is an algebraic integer.