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[itex]Q(\theta), \theta=\root 3\of{3}, \theta \in R [/itex]

Now, the inverse of a unit is the product of all of it's conjugates (possibly times -1), so if all of it's conjugates are in the field then it's inverse is also an algebraic integer in the ring of integers in question.

I would like to know if it works out so that the conjugates of units are _always_ in the field in question, so that units always have inverses in their ring of integers, even when all the conjugates of an arbitrary element don't.

If so, why?

Thanks.