# What does an algebraic integer have to be?

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.