For no good reason, I find it a surprising fact that there are algebraic integers of modulus that are not roots of unity. Clearly it is my intuition that needs adjusting, because as this paper by R. Daileda shows, such elements exist in any algebraic number field such that admits a complex field embedding.
Here is a version of Daileda’s argument that gives a stronger group theoretic result. Given an algebraic number field , let denote the set of units of the ring of algebraic integers , and let . If then we have
where is the complex conjugate of . It follows that
Either every unit of is a square, or has index in . In the first case , and in the second has index in .
Let . We apply Dirichlet’s Unit Theorem to and . By Dirichlet, the torsion-free part of has rank , where is the number of real conjugates of and is the number of pairs of complex conjugates; similarly, the torsion-free part of has rank . Hence
with equality if and only if is a torsion group, i.e. if and only if every algebraic integer in of modulus is a root of unity.
By applying the Tower Law to the chain of extensions we obtain
It follows that
The left-hand side is non-negative, and the right-hand side is non-positive. Hence every algebraic integer in of modulus is a root of unity if and only if , and . In particular, if then, as claimed, there are units in of modulus that are not roots of unity.
For example, let , let and let . Equivalently, since is generated by , we can think of as the splitting field of over . A laborious calculation with discriminants shows that . Since , there should be algebraic integers in of modulus that are not roots of unity. A brute force search shows that
is one such algebraic integer. Its minimum polynomial is
(A simpler real unit is .)
The answer by KcD to this question on Math.Stackexchange gives a similar example with a polynomial of degree , and explains the characteristic palindromic pattern of the polynomial above.
It is worth noting that the condition that has a complex embedding implies that complex conjugation is not in the centre of the Galois group of . Therefore is the smallest possible Galois group for a field extension containing algebraic integer units that do not have modulus .