I've always referred to that set as "algebraic numbers" ( https://en.wikipedia.org/wiki/Algebraic_number ). Since they are equipotent with integers, you _can_ call them that, but it's misleading.
No, algebraic integers are a different set than algebraic numbers. (A subset.)
Algebraic integers are much cooler, since there is a number theory on them: https://en.wikipedia.org/wiki/Algebraic_integer . (And also because the most basic facts about it, like that it forms a ring, are not trivial to prove, that's a good sign for a concept to be cool and useful.)
These two number sets are more or less in a relationship like regular integers (with a number theory), and rational numbers. In fact A = O/Z where A denotes the set of algebraic numbers, O denotes the set of algebraic integers, and Z denotes the set of integers.
As already mentioned by another poster, algebraic numbers are more general than algebraic integers, because the leading coefficient of the polynomial does not have to be one, similarly to the difference between rational numbers and integer numbers, where for the former the denominator does not have to be one, like for the latter.
Ahh, that would explain why the intersection of algebraic integers and Q is Z. I wasn’t convinced of that when I had the notion of algebraic numbers in place of algebraic integers.
I like teaching this kind of stuff to my grade 9 and 10 advanced math classes. It’s not that hard to understand and yet it gives students a sense of wonder about how math works. I might try to show the grade 10s algebraic integers now.
I've always referred to that set as "algebraic numbers" ( https://en.wikipedia.org/wiki/Algebraic_number ). Since they are equipotent with integers, you _can_ call them that, but it's misleading.