Examples of DVRs of residue char p and ramification e - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T18:16:46Z http://mathoverflow.net/feeds/question/43446 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/43446/examples-of-dvrs-of-residue-char-p-and-ramification-e Examples of DVRs of residue char p and ramification e Jeremy West 2010-10-24T23:47:56Z 2010-10-25T12:08:11Z <p>I am looking for concrete examples of a complete discrete valuation ring $R$ of characteristic 0, residue characteristic $p$ and ramification index $e$. By residue characteristic, I mean the characteristic of the field obtained by the quotient of $R$ with its unique maximal ideal $M$ and by ramification index I mean the largest positive integer $e$ such that $M^e \supseteq pR$. </p> <p>Without the restriction on the ramification, a simple example is the $p$-adic integers $\mathbb{Z}_p$. However, when we try to fix the ramification index, this becomes more challenging. For example, with $e = 2$ we can take $R = \mathbb{Z}_p[\sqrt{p}]$. The maximal ideal of this ring is $M = \sqrt{p}R$ which has ramification index 2.</p> <p>My question: is there a simple construction for such a ring with arbitrary $p$ and $e$? If not, can an infinite family of such rings be constructed that have a known ramification index $e > 2$?</p> http://mathoverflow.net/questions/43446/examples-of-dvrs-of-residue-char-p-and-ramification-e/43448#43448 Answer by Pete L. Clark for Examples of DVRs of residue char p and ramification e Pete L. Clark 2010-10-24T23:52:56Z 2010-10-24T23:52:56Z <p>The family of rings $\mathbb{Z}_p[p^{\frac{1}{e}}]$ does what you want. </p> <p>Because of your question, I gather you do not yet know the correspondence between totally ramified extensions and Eisenstein polynomials. For this see e.g. Serre's <em>Local Fields</em>, Lang's <em>Algebraic Number Theory</em>, or Section 4.3 of </p> <p><a href="http://math.uga.edu/~pete/8410Chapter4.pdf" rel="nofollow">http://math.uga.edu/~pete/8410Chapter4.pdf</a></p> http://mathoverflow.net/questions/43446/examples-of-dvrs-of-residue-char-p-and-ramification-e/43449#43449 Answer by Alex Bartel for Examples of DVRs of residue char p and ramification e Alex Bartel 2010-10-24T23:53:33Z 2010-10-25T01:54:09Z <p>Take any polynomial of degree $e$ that is Eisenstein at $p$, adjoin to $\mathbb{Q}_p$ a root of that polynomial and you will get a totally ramified extension of $\mathbb{Q}_p$ of degree $e$. Moreover, all totally ramified extensions of local fields arise in this way, by adjoining a root of an Eisenstein polynomial. So the ring of integers of that extension will be what you are looking for.</p> <p>Edit: I should have said that all the DVRs you obtain with the above procedure are of the form $\mathbb{Z}_p[\pi]$, where $\pi$ is a root of the Eisenstein polynomial you started with. See e.g. Serre, Local Fields, p 58.</p> http://mathoverflow.net/questions/43446/examples-of-dvrs-of-residue-char-p-and-ramification-e/43509#43509 Answer by Mikhail Bondarko for Examples of DVRs of residue char p and ramification e Mikhail Bondarko 2010-10-25T12:08:11Z 2010-10-25T12:08:11Z <p>If you fix a finite residue field, then there will be only a finite number of isomorphism classes of local fields with the given absolute ramification index. However, any (perfect) extension of a residue field yields an (unramified) extension with ramification index $1$, so unramified extensions of ${\mathbb Z}_p[p^{\frac{1/e}}]$ will give you an infinite family.</p>