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Q1: Do you known examples, where the different is not a principal ideal?

Q2: Is there a good interpretation for the reason, why this happens?

See e.g. Neukirch, Proposition 2.4, page 197.

The reason why I ask: the definition of the canonical additive character $\psi:x \mapsto \mathrm{e}^{2 \pi i (\mathrm{Tr}_{F / \mathbb{Q}} x \mod \mathbb{Z})}$ is somewhat unsatisfactory, if I want to identify the Pontryagin dual of the additive group of $\mathfrak{o}$ with the additive group of $F/\mathfrak{o}$, which simplifies some of notation involved when computing some $p$ adic integrals or Gauss sums.

Btw. with the canonical additive character, the Pontryagin duality is of the following form:

$$F / \mathfrak{D}^{-1} \cong \mathrm{Hom}_{} ( \mathfrak{o} , \mathbb{C}^\times).$$ where $\mathfrak{D}$ is the different and the isomorphism is given by $$ \xi \in F \mapsto \psi( \xi \cdotp).$$

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What do you mean by the definition of the canonical additive character? You didn't tell us what definition you are working with. (We don't all have Neukirch in front of us.) – KConrad Mar 28 '11 at 17:05
It is appropriate to recall the following beautiful result of Hecke: the ideal class of the different is always a square in the ideal class group. This is the last theorem in Weil's Basic Number Theory. – GH from MO Mar 28 '11 at 19:09
My previous comment shows that the different is a principal ideal whenever the ideal class group has exponent at most 2. – GH from MO Mar 28 '11 at 19:14
Are you interested in number fields or local fields? Your two questions are related to the different ideal in a number field, but when you say $\mathfrak o$ has Pontryagin dual $F/{\mathfrak o}$ then it sounds like you're talking about local fields. For an extension of local fields the different ideal is always principal since the ring of integers in the larger local field always has a single ring generator over the ring of integers in the smaller local field (at least if the extension is separable). – KConrad Mar 29 '11 at 1:44
up vote 10 down vote accepted

A partial answer:

Regarding Q1: An example for this is the number field generated by third root of $175$ . See e.g. a comment by KConrad on this question Which number fields are monogenic? and related questions

or also Ex 4.15 in these notes

which contain several explicit examples related to differents.

Regarding Q2: I can give at least a good reason when it does not happen (that is when it is principal). Namely, when the ring of integers is generated by a single element. If it is generated by a single element $u$, then the different is generated by $f'(u)$ where $f$ is the minimal polynomial of $u$.

A proof of this can be found in the above mentioned notes (Thm 4.3) as well as a discussion of the correct generalization of this result in the general case (Rem 4.5).

Thus, for example, for quadratic fields it is always principal. And, more generally, one does not have to look for examples for non-principal in too 'nice' fields (i.e., those where the ring of integers is generated by one element, such as cyclotomic fields).

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For completeness, the situation when the ring of integers is generated by a single element is Theorem 4.3 in the .pdf file you link to in your answer. – KConrad Mar 28 '11 at 17:14
@KConrad, thank you, I will add this. – user9072 Mar 28 '11 at 23:21

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