This question is motivated by a Quora post and the top answer to it.
The post wishes to make explicit the difference between units in cyclotomic fields and cyclotomic units.
One problem with answering such a question is the difficulty in computing fundamental units in cyclotomic fields or subfields thereof.
However, there is a case when we know the fundamental unit---real quadratic fields.
This led me to the answer to a question on MO which looks at whether the norm of a cyclotomic unit is a fundamental unit in a real quadratic field. The answer is given in the negative.
However, while checking whether this answers the original question, one notices the gap: it is not clear that the cyclotomic units contained in the real quadratic field are norms from the cyclotomic field that contain this field.
In summary, here are the questions:
Let $K$ be a real quadratic field and $E$ be a cyclotomic field that contains it. Let $C_E$ be the group of (i.e. generated by) cyclotomic units in $E$ and $C_K$ be the intersection of $C_E$ with $K$.
Is there a formula for the generator of $C_K$?
Is there a situation where this generator is not the norm of a cyclotomic unit from $C_E$? (If not, then (1) is already answered!)
What is the smallest $K$ for which the fundamental unit is not in $C_K$?
In (3), I am guessing that there is always such a $K$ and so it would answer the original question. If not:
- What is the smallest Abelian extension of $\mathbb{Q}$ in which one can give an example of a unit which is not a cyclotomic unit?
