On Galois Groups of Local Fields
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APPLICATIONS of GALOIS THEORY 1. Finite Fields Let F Be a Finite Field
CHAPTER IX APPLICATIONS OF GALOIS THEORY 1. Finite Fields Let F be a finite field. It is necessarily of nonzero characteristic p and its prime field is the field with p r elements Fp.SinceFis a vector space over Fp,itmusthaveq=p elements where r =[F :Fp]. More generally, if E ⊇ F are both finite, then E has qd elements where d =[E:F]. As we mentioned earlier, the multiplicative group F ∗ of F is cyclic (because it is a finite subgroup of the multiplicative group of a field), and clearly its order is q − 1. Hence each non-zero element of F is a root of the polynomial Xq−1 − 1. Since 0 is the only root of the polynomial X, it follows that the q elements of F are roots of the polynomial Xq − X = X(Xq−1 − 1). Hence, that polynomial is separable and F consists of the set of its roots. (You can also see that it must be separable by finding its derivative which is −1.) We q may now conclude that the finite field F is the splitting field over Fp of the separable polynomial X − X where q = |F |. In particular, it is unique up to isomorphism. We have proved the first part of the following result. Proposition. Let p be a prime. For each q = pr, there is a unique (up to isomorphism) finite field F with |F | = q. Proof. We have already proved the uniqueness. Suppose q = pr, and consider the polynomial Xq − X ∈ Fp[X]. As mentioned above Df(X)=−1sof(X) cannot have any repeated roots in any extension, i.e. -
Finitely Generated Pro-P Galois Groups of P-Henselian Fields
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF PURE AND APPLIED ALGEBRA ELSEYIER Journal of Pure and Applied Algebra 138 (1999) 215-228 Finitely generated pro-p Galois groups of p-Henselian fields Ido Efrat * Depurtment of Mathematics and Computer Science, Ben Gurion Uniaersity of’ the Negev, P.O. BO.Y 653. Be’er-Sheva 84105, Israel Communicated by M.-F. Roy; received 25 January 1997; received in revised form 22 July 1997 Abstract Let p be a prime number, let K be a field of characteristic 0 containing a primitive root of unity of order p. Also let u be a p-henselian (Krull) valuation on K with residue characteristic p. We determine the structure of the maximal pro-p Galois group GK(P) of K, provided that it is finitely generated. This extends classical results of DemuSkin, Serre and Labute. @ 1999 Elsevier Science B.V. All rights reserved. A MS Clussijicution: Primary 123 10; secondary 12F10, 11 S20 0. Introduction Fix a prime number p. Given a field K let K(p) be the composite of all finite Galois extensions of K of p-power order and let GK(~) = Gal(K(p)/K) be the maximal pro-p Galois group of K. When K is a finite extension of Q, containing the roots of unity of order p the group GK(P) is generated (as a pro-p group) by [K : Cl!,,]+ 2 elements subject to one relation, which has been completely determined by Demuskin [3, 41, Serre [20] and Labute [14]. -
Determining the Galois Group of a Rational Polynomial
Determining the Galois group of a rational polynomial Alexander Hulpke Department of Mathematics Colorado State University Fort Collins, CO, 80523 [email protected] http://www.math.colostate.edu/˜hulpke JAH 1 The Task Let f Q[x] be an irreducible polynomial of degree n. 2 Then G = Gal(f) = Gal(L; Q) with L the splitting field of f over Q. Problem: Given f, determine G. WLOG: f monic, integer coefficients. JAH 2 Field Automorphisms If we want to represent automorphims explicitly, we have to represent the splitting field L For example as splitting field of a polynomial g Q[x]. 2 The factors of g over L correspond to the elements of G. Note: If deg(f) = n then deg(g) n!. ≤ In practice this degree is too big. JAH 3 Permutation type of the Galois Group Gal(f) permutes the roots α1; : : : ; αn of f: faithfully (L = Spl(f) = Q(α ; α ; : : : ; α ). • 1 2 n transitively ( (x α ) is a factor of f). • Y − i i I 2 This action gives an embedding G S . The field Q(α ) corresponds ≤ n 1 to the subgroup StabG(1). Arrangement of roots corresponds to conjugacy in Sn. We want to determine the Sn-class of G. JAH 4 Assumption We can calculate “everything” about Sn. n is small (n 20) • ≤ Can use table of transitive groups (classified up to degree 30) • We can approximate the roots of f (numerically and p-adically) JAH 5 Reduction at a prime Let p be a prime that does not divide the discriminant of f (i.e. -
Rolle's Theorem Over Local Fields
Rolle's Theorem over Local Fields Cristina M. Ballantine Thomas R. Shemanske April 11, 2002 Abstract In this paper we show that no non-archimedean local field has Rolle's property. 1 Introduction Rolle's property for a field K is that if f is a polynomial in K[x] which splits over K, then its derivative splits over K. This property is implied by the usual Rolle's theorem taught in calculus for functions over the real numbers, however for fields with no ordering, it is the best one can hope for. Of course, Rolle's property holds not only for the real numbers, but also for any algebraically- or real-closed field. Kaplansky ([3], p. 30) asks for a characterization of all such fields. For finite fields, Rolle's property holds only for the fields with 2 and 4 elements [2], [1]. In this paper, we show that Rolle's property fails to hold over any non-archimedean local field (with a finite residue class field). In particular, such fields include the completion of any global field of number theory with respect to a nontrivial non-archimedean valuation. Moreover, we show that there are counterexamples for Rolle's property for polynomials of lowest possible degree, namely cubics. 2 Rolle's Theorem Theorem 2.1. Rolle's property fails to hold over any non-archimedean local field having finite residue class field. Proof. Let K be a non-archimedean local field with finite residue class field. Let O be the ring of integers of K, and P its maximal ideal. -
Local-Global Methods in Algebraic Number Theory
LOCAL-GLOBAL METHODS IN ALGEBRAIC NUMBER THEORY ZACHARY KIRSCHE Abstract. This paper seeks to develop the key ideas behind some local-global methods in algebraic number theory. To this end, we first develop the theory of local fields associated to an algebraic number field. We then describe the Hilbert reciprocity law and show how it can be used to develop a proof of the classical Hasse-Minkowski theorem about quadratic forms over algebraic number fields. We also discuss the ramification theory of places and develop the theory of quaternion algebras to show how local-global methods can also be applied in this case. Contents 1. Local fields 1 1.1. Absolute values and completions 2 1.2. Classifying absolute values 3 1.3. Global fields 4 2. The p-adic numbers 5 2.1. The Chevalley-Warning theorem 5 2.2. The p-adic integers 6 2.3. Hensel's lemma 7 3. The Hasse-Minkowski theorem 8 3.1. The Hilbert symbol 8 3.2. The Hasse-Minkowski theorem 9 3.3. Applications and further results 9 4. Other local-global principles 10 4.1. The ramification theory of places 10 4.2. Quaternion algebras 12 Acknowledgments 13 References 13 1. Local fields In this section, we will develop the theory of local fields. We will first introduce local fields in the special case of algebraic number fields. This special case will be the main focus of the remainder of the paper, though at the end of this section we will include some remarks about more general global fields and connections to algebraic geometry. -
An Invitation to Local Fields
An Invitation to Local Fields Groups, Rings and Group Rings Ubatuba-S˜ao Paulo, 2008 Eduardo Tengan (ICMC-USP) “To get a book from these texts, only scissors and glue were needed.” J.-P. Serre, in response to receiving the 1995 Steele Prize for his book “Cours d’Arithm´etique” Copyright c 2008 E. Tengan Permission is granted to make and distribute verbatim copies of this document provided the copyright notice and this permission notice are preserved on all copies. The author was supported by FAPESP grant 06/59613-8. Preface 1 What is a Local Field? Historically, the first local field, the field of p-adic numbers Qp, was introduced in 1897 by Kurt Hensel, in an attempt to borrow ideas and techniques of power series in order to solve problems in Number Theory. Since its inception, local fields have attracted the attention of several mathematicians, and have found innumerable applications not only to Number Theory but also to Representation Theory, Division Algebras, Quadratic Forms and Algebraic Geometry. As a result, local fields are now consolidated as part of the standard repertoire of contemporary Mathematics. But what exactly is a local field? Local field is the name given to any finite field extension of either the field of p-adic numbers Qp or the field of Laurent power series Fp((t)). Local fields are complete topological fields, and as such are not too distant relatives of R and C. Unlike Q or Fp(t) (which are global fields), local fields admit a single valuation, hence the tag ‘local’. Local fields usually pop up as completions of a global field (with respect to one of the valuations of the latter). -
ON R-EXTENSIONS of ALGEBRAIC NUMBER FIELDS Let P Be a Prime
ON r-EXTENSIONS OF ALGEBRAIC NUMBER FIELDS KENKICHI IWASAWA1 Let p be a prime number. We call a Galois extension L of a field K a T-extension when its Galois group is topologically isomorphic with the additive group of £-adic integers. The purpose of the present paper is to study arithmetic properties of such a T-extension L over a finite algebraic number field K. We consider, namely, the maximal unramified abelian ^-extension M over L and study the structure of the Galois group G(M/L) of the extension M/L. Using the result thus obtained for the group G(M/L)> we then define two invariants l(L/K) and m(L/K)} and show that these invariants can be also de termined from a simple formula which gives the exponents of the ^-powers in the class numbers of the intermediate fields of K and L. Thus, giving a relation between the structure of the Galois group of M/L and the class numbers of the subfields of L, our result may be regarded, in a sense, as an analogue, for L, of the well-known theorem in classical class field theory which states that the class number of a finite algebraic number field is equal to the degree of the maximal unramified abelian extension over that field. An outline of the paper is as follows: in §1—§5, we study the struc ture of what we call T-finite modules and find, in particular, invari ants of such modules which are similar to the invariants of finite abelian groups. -
Local Fields
Part III | Local Fields Based on lectures by H. C. Johansson Notes taken by Dexter Chua Michaelmas 2016 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures. They are nowhere near accurate representations of what was actually lectured, and in particular, all errors are almost surely mine. The p-adic numbers Qp (where p is any prime) were invented by Hensel in the late 19th century, with a view to introduce function-theoretic methods into number theory. They are formed by completing Q with respect to the p-adic absolute value j − jp , defined −n n for non-zero x 2 Q by jxjp = p , where x = p a=b with a; b; n 2 Z and a and b are coprime to p. The p-adic absolute value allows one to study congruences modulo all powers of p simultaneously, using analytic methods. The concept of a local field is an abstraction of the field Qp, and the theory involves an interesting blend of algebra and analysis. Local fields provide a natural tool to attack many number-theoretic problems, and they are ubiquitous in modern algebraic number theory and arithmetic geometry. Topics likely to be covered include: The p-adic numbers. Local fields and their structure. Finite extensions, Galois theory and basic ramification theory. Polynomial equations; Hensel's Lemma, Newton polygons. Continuous functions on the p-adic integers, Mahler's Theorem. Local class field theory (time permitting). Pre-requisites Basic algebra, including Galois theory, and basic concepts from point set topology and metric spaces. -
Number Theory
Number Theory Alexander Paulin October 25, 2010 Lecture 1 What is Number Theory Number Theory is one of the oldest and deepest Mathematical disciplines. In the broadest possible sense Number Theory is the study of the arithmetic properties of Z, the integers. Z is the canonical ring. It structure as a group under addition is very simple: it is the infinite cyclic group. The mystery of Z is its structure as a monoid under multiplication and the way these two structure coalesce. As a monoid we can reduce the study of Z to that of understanding prime numbers via the following 2000 year old theorem. Theorem. Every positive integer can be written as a product of prime numbers. Moreover this product is unique up to ordering. This is 2000 year old theorem is the Fundamental Theorem of Arithmetic. In modern language this is the statement that Z is a unique factorization domain (UFD). Another deep fact, due to Euclid, is that there are infinitely many primes. As a monoid therefore Z is fairly easy to understand - the free commutative monoid with countably infinitely many generators cross the cyclic group of order 2. The point is that in isolation addition and multiplication are easy, but together when have vast hidden depth. At this point we are faced with two potential avenues of study: analytic versus algebraic. By analytic I questions like trying to understand the distribution of the primes throughout Z. By algebraic I mean understanding the structure of Z as a monoid and as an abelian group and how they interact. -
Techniques for the Computation of Galois Groups
Techniques for the Computation of Galois Groups Alexander Hulpke School of Mathematical and Computational Sciences, The University of St. Andrews, The North Haugh, St. Andrews, Fife KY16 9SS, United Kingdom [email protected] Abstract. This note surveys recent developments in the problem of computing Galois groups. Galois theory stands at the cradle of modern algebra and interacts with many areas of mathematics. The problem of determining Galois groups therefore is of interest not only from the point of view of number theory (for example see the article [39] in this volume), but leads to many questions in other areas of mathematics. An example is its application in computer algebra when simplifying radical expressions [32]. Not surprisingly, this task has been considered in works from number theory, group theory and algebraic geometry. In this note I shall give an overview of methods currently used. While the techniques used for the identification of Galois groups were known already in the last century [26], the involved calculations made it almost imprac- tical to do computations beyond trivial examples. Thus the problem was only taken up again in the last 25 years with the advent of computers. In this note we will restrict ourselves to the case of the base field Q. Most methods generalize to other fields like Q(t), Qp, IFp(t) or number fields. The results presented here are the work of many mathematicians. I tried to give credit by references wherever possible. 1 Introduction We are given an irreducible polynomial f Q[x] of degree n and asked to determine the Galois group of its splitting field2 L = Spl(f). -
24 Local Class Field Theory
18.785 Number theory I Fall 2015 Lecture #24 12/8/2015 24 Local class field theory In this lecture we give a brief overview of local class field theory, following [1, Ch. 1]. Recall that a local field is a locally compact field whose topology is induced by a nontrivial absolute value (Definition 9.1). As we proved in Theorem 9.7, every local field is isomorphic to one of the following: • R or C (archimedean); • finite extension of Qp (nonarchimedean, characteristic 0); • finite extension of Fp((t)) (nonarchimedean, characteristic p). In the nonarchimedean cases, the ring of integers of a local field is a complete DVR with finite residue field. The goal of local class field theory is to classify all finite abelian extensions of a given local field K. Rather than considering each finite abelian extension L=K individually, we will treat them all at once, by fixing once and for all a separable closure Ksep of K and working in the maximal abelian extension of K inside Ksep. Definition 24.1. Let K be field with separable closure Ksep. The field [ Kab := L L ⊆ Ksep L=K finite abelian is the maximal abelian extension of K (in Ksep). The field Kab contains the field Kunr, the maximal unramified subextension of Ksep=K; this is obvious in the archimedean case, and in the nonarchimedean case it follows from Theorem 10.1, which implies that Kunr is isomorphic to the algebraic closure of the residue field of K, which is abelian because the residue field is finite every algebraic extension of a finite field is pro-cyclic, and in particular, abelian. -
The Orthogonal U-Invariant of a Quaternion Algebra
The orthogonal u-invariant of a quaternion algebra Karim Johannes Becher, Mohammad G. Mahmoudi Abstract In quadratic form theory over fields, a much studied field invariant is the u-invariant, defined as the supremum over the dimensions of anisotropic quadratic forms over the field. We investigate the corre- sponding notions of u-invariant for hermitian and for skew-hermitian forms over a division algebra with involution, with a special focus on skew-hermitian forms over a quaternion algebra. Under certain condi- tions on the center of the quaternion algebra, we obtain sharp bounds for this invariant. Classification (MSC 2000): 11E04, 11E39, 11E81 Key words: hermitian form, involution, division algebra, isotropy, system of quadratic forms, discriminant, Tsen-Lang Theory, Kneser’s Theorem, local field, Kaplansky field 1 Involutions and hermitian forms Throughout this article K denotes a field of characteristic different from 2 and K× its multiplicative group. We shall employ standard terminology from quadratic form theory, as used in [9]. We say that K is real if K admits a field ordering, nonreal otherwise. By the Artin-Schreier Theorem, K is real if and only if 1 is not a sum of squares in K. − Let ∆ be a division ring whose center is K and with dimK(∆) < ; we refer to such ∆ as a division algebra over K, for short. We further assume∞ that ∆ is endowed with an involution σ, that is, a map σ : ∆ ∆ such that → σ(a + b)= σ(a)+ σ(b) and σ(ab)= σ(b)σ(a) hold for any a, b ∆ and such ∈ 1 that σ σ = id∆.