Algebraic integer

Question

Can we drop the condition monic in the definition of an algebraic integer? My answer is no because this would imply that $1/2$ is algebraic integer being the root of $2x-1$. I am looking for a better argument without using an example.

Answer 1

There is no better answer. A definition of algebraic integer is: a complex number which is a root of a monic polynomial with integer coefficients. A definition of algebraic number is: a complex number which is a root of a non-null polynomial with integer coefficients. The simplest and most direct way of showing that these two concepts are not the same consists in finding an algebraic number which is not an algebraic integer. And $\frac12$ is a very simple example of such a number.

Answer 2

Demonstrating counterexamples may not seem like "profound" proofs, but they don't have to be. Their job is often to disprove potentially profound results, and as such, for want of a better word, they are "anti-profound".

Of course, there are trivial counterexamples and there are interesting counterexamples, and of the two the latter are more ... interesting. This is because trivial counterexamples can sometimes be fixed by adding some simple constraint to the theorem (adding "non-empty" to a set, for instance).

In this case I would say that $\frac12$ is a good counterexample, trivial though it may seem, because it doesn't exploit some small technicality but instead demonstrates the exact thing that separates the two concepts.