Achilles and the tortoise

Hello there! These are notes for the fifteenth class of the course " Topics in algebra and number theory " held in Block 4 of the academic year 2017/18 at the University of Copenhagen . In the previous lecture we have defined the ramification filtration of a finite, Galois extension of complete, discretely valued fields. This filtration gives us a way of studying the Galois group of any extension of complete, discretely valued fields. In this lecture we will concentrate on Galois extensions of a local field that are abelian , i.e. the Galois group is commutative. In particular we will see that every (tamely ramified) abelian extension of \( \mathbb{Q}_{p} \) is contained in a cyclotomic extension. In order to do so, we will need a lemma which will enable us to kill the ramification in an extension of valued fields after changing the base field: this is the so-called Abhyankhar's lemma .

Hello there! These are notes for the fourteenth class of the course " Topics in algebra and number theory " held in Block 4 of the academic year 2017/18 at the University of Copenhagen . In the previous lecture we have seen how every extension \( (K,\phi) \hookrightarrow (L,\psi) \) of complete, non-Archimedean, discretely valued fields can be split into an unramified and a totally ramified extension. We have seen moreover that unramified extensions correspond bijectively to separable extensions of the residue field and totally ramified extensions correspond to Eisenstein polynomials. In this lecture we will see how the presence of the absolute values allows us to define a filtration on the Galois group of any Galois extension \( (K,\phi) \hookrightarrow (L,\psi) \) of complete, non-Archimedean, discretely valued fields. This filtration will be really useful in the following lectures, to prove the theorem of Kronecker and Weber.

Hello there! These are notes for the thirteenth class of the course " Topics in algebra and number theory " held in Block 4 of the academic year 2017/18 at the University of Copenhagen . In the previous lecture we discussed extensions of absolute values. We have seen in particular that for every valued field \( (K,\phi) \) and every finite, separable extension \( K \subseteq L \) there exist only a finite number of absolute values \( \psi \colon L \to \mathbb{R}_{\geq 0} \) which extend \( \phi \). Given such an absolute value \( \psi \) we have also defined the ramification index \( e(\psi \mid \phi) \) and the inertia index \( f(\psi \mid \phi) \) which measure how "close" the two absolute values are, in two different ways. In this lecture we will concentrate on extensions of complete, discretely valued fields which are either unramified , i.e. \( e(\psi \mid \phi) = 1 \) or totally ramified , i.e.\( f(\psi \mid \phi) = 1 \). We will prove that all the exte...

Hello there! These are notes for the twelfth class of the course " Topics in algebra and number theory " held in Block 4 of the academic year 2017/18 at the University of Copenhagen . In the previous lecture we have seen two useful applications of completeness: Hensel's lemma , which allows us to solve equations in complete fields by solving them in their residue fields, and a characterization of norms on finite dimensional vector spaces over complete fields. The aim of this lecture is to use the second result as a starting point to study the theory of the extensions of absolute values to an algebraic extension. We have already seen in the second lecture that if we have a field extension \( K \hookrightarrow L \) and an absolute value \( \psi \colon L \to \mathbb{R}_{\geq 0} \) we can restrict it to an absolute value \( \phi \colon K \to \mathbb{R}_{\geq 0} \). Suppose now that we have an absolute value \( \phi \colon K \to \mathbb{R}_{\geq 0} \). Can we exten...

Hello there! These are notes for the eleventh class of the course " Topics in algebra and number theory " held in Block 4 of the academic year 2017/18 at the University of Copenhagen . In the past lectures we have seen a lot of complete fields. In particular, we have seen in the fourth lecture that every complete Archimedean field is isomorphic to a finite extension of \( \mathbb{R} \), and we have seen in the ninth lecture that every non-Archimedean complete field which is discretely valued and whose residue field is finite is either a finite extension of \( \mathbb{Q}_p \) or a finite extension of \( \mathbb{F}_p((T)) \). We have seen moreover in the tenth lecture that we can describe these fields as fields of Laurent series in \( p \) and in \( T \) respectively (with the important difference that in \( \mathbb{Q}_p \) we have to sum with carrying ) or as fraction fields of inverse limits of finite rings.

Hello there! These are notes for the ninth class of the course " Topics in algebra and number theory " held in Block 4 of the academic year 2017/18 at the University of Copenhagen . In the previous lecture we have given a complete characterization of non-Archimedean fields which are locally compact . In particular, we have seen that they are all finite extensions of \( \mathbb{Q}_p \) or \( \mathbb{F}_p((T)) \). Today we are going to study more these two fields, giving an alternative description of them which uses inverse limits .

Hello there! These are notes for the ninth class of the course " Topics in algebra and number theory " held in Block 4 of the academic year 2017/18 at the University of Copenhagen . In the previous lectures we have analyzed the set \( \Sigma_K \) of places of a field \( K \), and we have completely characterized it when \( K \) is a number field. In order to do so we defined in the fourth lecture the notion of completion of a valued field \( (K,\phi) \) and we have seen that every complete, Archimedean field is isomorphic to \( (\mathbb{R},\lvert \cdot \rvert) \) or to \( (\mathbb{C},\lVert \cdot \rVert) \). So, what about the non-Archimedean case? Do we have a similar classification result? The answer is a resounding no! More precisely, we have an infinite number of non-Archimedean complete fields which are not isomorphic, as we will see by the end of the lecture.