# What is the real subring of a ring of cyclotomic integers?

I am looking at tilings whose vertices lie in a ring of cyclotomic integers. These tilings are of interest as they can have interesting scaling properties or be substitution tilings. Interesting scalings are given by elements of the real subring, especially the units. I have two related questions:

1) What is the real subring of the ring of cyclotomic integers of order n?

2) What are the generators of the group of units of these real subrings?

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For your first question, it is generated by what you'd expect (?), namely $\zeta_n + \zeta_n^{-1}$. I believe this is discussed in the first chapter of Washington's book on cyclotomic fields. For the second question, there is generally no formula for a set of generators of the unit group, but there is a formula for an explicit set of generators of a subgroup of finite index (called circular units). Again, this is in Washington's book. – KConrad Jul 10 '13 at 2:17
Thank you, it is good to confirm that the obvious answer does generate the subring. Sadly for the second part the circular units are not so interesting as I again wish to consider the real subring. – Edmund Harriss Jul 10 '13 at 2:38