Answer to Question #312751 in Algebra for deepu

Question

prove that the product of two algebraic integers is an algebraic integers

Answer 1

there are monic polynomials f,g such that f(a)=0 and g(b)=0. Wolog g(0)≠0. If it was, the whole polynomial could be divided by the variable and would still fulfill g(b)=0.

Let A be the companion matrix of f and B the companion matrix of g. B is invertible because g(0)≠0. One of the eigenvalues of A is a, and one of the eigenvalues of B is b, which means that one of the eigenvalues of B−1 is b−1.

Now set C=A⊗B−1 (Kronecker product). Then one of the eigenvalues of C is ab−1, which means that ab−1 is one of the roots of the characteristic polynomial χC. This makes ab−1 algebraic over F, because all of the elements of C are in F. Therefore, all coefficients of χC are in F, too.

Credit: https://math.stackexchange.com/questions/3256143/if-a-and-b-are-algebraic-then-frac-ab-is-algebraic/3256282