@deilann
Bouncing back on the subject, I took Artin's #GaloisTheory with me to #Benasque and I think on page 25 there is a complete discussion. As you said any #algebraicNumber, defined as a root of a real polynomial, can be seen as a root of a unique polynomial of lowest degree which is irreducible, i.e. not divisible by any other poly omial with coefficients in the base field.

The coefficient of the highest power needs to be 1, one should divide the contending polynomials by each other and evaluate at the shared root showing that, having the same degree, they have 0 polynomial ring division remainder so must be the same.