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I would like to understand why the sum and product of algebraic integers are algebraic integers.

For algebraic numbers (not integers) there is the wonderful website https://www.dpmms.cam.ac.uk/~wtg10/galois.html which uses only basic linear algebra. A short version of that is in this MSE-answer: https://math.stackexchange.com/a/155153/564656

My Question:

Can this method or something similar be used for algebraic integers?

Ben Grossmann
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L. Milla
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Yes, the exact same proof (at least, the brief MSE version) shows that the sum/product of algebraic integers is algebraic.

As in the linked post, take $V = F[x,y]/(p(x),q(y))$. Verify that because $p$ is monic, the matrix of the operator $\alpha(x,y) \mapsto x\,\alpha(x,y)$ has integer coefficients. The same holds for $\alpha(x,y) \mapsto y\,\alpha(x,y)$.

Now, the sum/product of matrices with integer coefficients is also matrix with integer coefficients. So, the matrices associated with $x + y,xy$ have integer coefficients. By the Cayley Hamilton theorem, $x + y$ and $xy$ therefore satisfy a monic polynomial with integer coefficients.

Ben Grossmann
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  • thank you! But with Cayley-Hamilton you bring in quite the big gun... Is it also possible with "smaller" requirements? – L. Milla Mar 24 '20 at 11:10
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    @L.Milla You could explicitly find the relation for $x + y$ and $xy$, which is more work but avoids the Cayley Hamilton theorem. In particular, it's easy once you show that the matrices for $xy$ and $x + y$ are the Kronecker product and sum of the matrices associated with $x$ and $y$ (as Qiaochu comments on his answer) – Ben Grossmann Mar 29 '20 at 19:10