# 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.

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).