When is the different in a number field a principal ideal? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T10:24:35Z http://mathoverflow.net/feeds/question/59823 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/59823/when-is-the-different-in-a-number-field-a-principal-ideal When is the different in a number field a principal ideal? Marc Palm 2011-03-28T12:20:40Z 2011-03-29T05:38:54Z <p>Q1: Do you known examples, where the different is not a principal ideal?</p> <p>Q2: Is there a good interpretation for the reason, why this happens?</p> <p>See e.g. Neukirch, Proposition 2.4, page 197.</p> <p>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.</p> <p>Btw. with the canonical additive character, the Pontryagin duality is of the following form:</p> <p>$$F / \mathfrak{D}^{-1} \cong \mathrm{Hom}_{ab.group} ( \mathfrak{o} , \mathbb{C}^\times).$$ where $\mathfrak{D}$ is the different and the isomorphism is given by $$\xi \in F \mapsto \psi( \xi \cdotp).$$</p> http://mathoverflow.net/questions/59823/when-is-the-different-in-a-number-field-a-principal-ideal/59831#59831 Answer by quid for When is the different in a number field a principal ideal? quid 2011-03-28T13:15:11Z 2011-03-28T23:26:41Z <p>A partial answer:</p> <p>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 <a href="http://mathoverflow.net/questions/21267/which-number-fields-are-monogenic-and-related-questions" rel="nofollow">http://mathoverflow.net/questions/21267/which-number-fields-are-monogenic-and-related-questions</a> </p> <p>or also Ex 4.15 in these notes </p> <p><a href="http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/different.pdf" rel="nofollow">http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/different.pdf</a></p> <p>which contain several explicit examples related to differents.</p> <p>Regarding Q2: I can give at least a good reason when it does <em>not</em> 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 $u$. </p> <p>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). </p> <p>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). </p>