Consider a set of algebraic integers which lie on the unit circle, they will generate a multiplicative subgroup of $\mathbb S^1$. These objects do they have a name?

I would guess they contain useful arithmetic/number theoretic information, for example if the elements of the generating subset are the roots of a polynomial, is it really so? What kind of information do they contain?

Has the group structure of the units of an extension field which lie on the unit circle $\mathbb S^1$ and the subgroup of it which is made up of algebraic integers been used to obtain results which can be used in actual application.

Would greatly appreciate if you could suggest a reference.

Regards Vagabond

PS

It would be real nice if you could answer keeping in mind that I do not know much Algebra/ commutative algebra/ algebraic geometry. But I hope that does not stop you from answering.

Consider a set of algebraic integers which lie on the unit circle, they will generate a multiplicative subgroup of $\mathbb S^1$. Do these objects have a name?

I would guess they contain useful arithmetic/number theoretic information, for example if the generating set is the set of roots of an irreducible polynomial, what kind of information would they contain?

Has the group structure of the elements of a number field which lie on the unit circle $\mathbb S^1$ and the subgroup of it which is made up of algebraic integers been studied.

Would greatly appreciate if you could suggest a reference.

Regards Vagabond

PS

It would be real nice if you could answer keeping in mind that I do not know much Algebra/ commutative algebra/ algebraic geometry. But I hope that does not stop you from answering.