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 coefficientsÂ (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.*

As you said algebraic concepts will be obtained with this integer. Keep on posting detailed concept of polynomial integration. Thanks for this post