What does an integer have to be?

• No matter how you extend \(\mathcal{Q}\), the integers which lie in \(\mathcal{Q}\) must lie in \(\mathcal{Z}\).
• If \(\alpha\) is an integer, then so are its conjugates.
• The sums and products of integers are also integers.

From this we may describe what an algebraic integer must be.

Start with a root \(\alpha\).

Look at all of it’s conjugates. \(\alpha, \alpha’, \alpha”, …\) By conjugates, I mean the elements that have the same minimal polynomial as \(\alpha\) (that is, the elements that cannot be distinguished).

Look at all products and sums of \(\alpha, \alpha’, \alpha”, …\).

Look at symmetric polynomials in \(\alpha, \alpha’, \alpha”, …\). Things that are symmetric in the roots must have quadratic coefficients (by the fundamental theorem of Galois theory wrt symmetric polynomials), and it must be integral because sums/products of integral things must be integral. So, by Vieta, the minimal polynomial must be monic.

Thanks to Aaron Slipper and Hecke.