ON THE GALOIS GROUPS OF THE EXPONENTIAL TAYLOR POLYNOMIALS

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1 ON THE GALOIS GROUPS OF THE EXPONENTIAL TAYLOR POLYNOMIALS Autor(e): Colema, Robert F Objekttyp: Article Zeitschrift: L'Eseigemet Mathématique Bad(Jahr): 33(1987) Heft 1-2: L'ENSEIGNEMENT MATHÉMATIQUE Erstellt am: Mar 26, 2014 Persisteter Lik: Nutzugsbediguge Mit dem Zugriff auf de vorliegede Ihalt gelte die Nutzugsbediguge als akzeptiert Die agebotee Dokumete stehe für icht-kommerzielle Zwecke i Lehre, Forschug ud für die private Nutzug frei zur Verfügug Eizele Dateie oder Ausdrucke aus diesem Agebot köe zusamme mit diese Nutzugsbediguge ud uter dere Eihaltug weitergegebe werde Die Speicherug vo Teile des elektroische Agebots auf adere Server ist ur mit vorheriger schriftlicher Geehmigug möglich Die Rechte für diese ud adere Nutzugsarte der Ihalte liege beim Herausgeber bzw beim Verlag Ei Diest der ETH-Bibliothek Rämistrasse 101, 8092 Zürich, Schweiz retro@sealsch

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3 We shall use 1 for A ad B This will imply G is trasitive ad together with 2 ad 3 will imply G cotais A for 8 We shall use the differetial équatio satisfied by the expoetial fuctio ad 2 agai to perform C Fially, we shall use 4 to complète the proof (We shall also require libéral doses of Galois theory) I Review of the Newto Polygo Let be a polyomial over Q p Cosider the poits : i the Cartesia plae The Newto polygo of g is defied to be the lower covex hull of thèse poits Example The Newto Polygo of f 7 (x) cosidered over Q 2 is, The mai theorem about thèse polygos is Theorem NP Let (x 0, y0),y 0 ), (x l9 j/j,, (x p, y p ) déote the successive vertices of this polygo The over Q p, g factors as follows :

4 where the degree of g t is x i -x f, i i + fyt-yi-i\ O hâve vaïuatw P \Xi-Xi-J We call the ratioal umbers,, Xf %i-i i^ 1 ad ail the roots of g t {x) i the slopes of g Example The polyomial /7/ 7 has three factors over Q 2, of degrees 4, 2 ad 1, respectively, which hâve slopes 3/4, 1/2 ad 0 Corollary Let d be a positive iteger Suppose that d divides the deomiator of each slope (i lowest terms) of g The d divides the degree of each factor of g over Q p Proof It suffices to show that d divides the degree of each irreducible factor of g Let h be such a factor Let aeqp be a root of h Sice d divides the deomiator of the valuatio of a (by Theorem NP), it follows that d divides the idex of ramificatio of the extesio Qp(ol)/Q p (ol)/q p divides the degree of the extesio which equals the degree of h which II Applicatio to the Expoetial Taylor Polyomials Fix a prime umber p Lemma Suppose k is a positive iteger ad where 0 a t <p The This is easy ad well kow Now write where 1 > 2 2 > > s ad 0 < b t < p Let

5 Lemma The vertices of the Newto polygo of f are This follows easily from the previous lemmas It follows that the slopes of / are Corollary A Suppose that p m divides The p m divides the degree of each factor of f over Q p Proof Sice p m divides, m s < s _ x < Hece, it follows from (1) that p m divides the deomiator of each m Therefore the corollary follows from the corollary to Theorem NP Corollary B Suppose that p k The p k divides the degree of the splittig field of f over Q p Proof The hypothèses imply that k< l miatorof mlm k 1 Hece p divides the deo 1 As above this implies that p k divides the degree of ay extesio of Q p formed by adjoiig a root of / with valuatio mlm 1 This yields the corollary III Global Coclusios A ad B A f is irreducible Suppose is the prime factorizatio of Corollary A implies that, for each prime p, p Hp divides the degree of each factor of / over Q The coclusio follows B Suppose /2 < p< is a prime umber The G cotais a p-cycle By Corollary B, p divides the degree of the splittig field of / over Q p which divides the degree of the splittig field of / over Q Hece p divides the order of G By Cauchy's Theorem G cotais a élémet of order p The coclusio follows sice the oly élémets of order p i S are p-cycles if p > /2

6 1 Hilbert [H] proved that there exist extesios of Q with Galois group S IV CALCULATION OF THE DISCRIMINANT V Ed of Proof Tschebyshev's Theorem (2 above) implies that for each > 1 there is a prime umber p such that Hece D is ot a square if is odd If = 2(4) the DRD R <0 so it is ot a square i this case either Fially if 4 divides the we see that D is a square This complètes the proof for 8 The remaiig cases ca be hadled idividually usig the above results ad facts about S for small (See [S-2]) VI Fial Remarks The splittig fields of the expoetial polyomials provide explicit examples of such extesios Moreover, they provide examples of such extesios ramified oly at the primes dividig the order of the Galois group, a property ot predictable by Hilbert's methods (I fact, as ca easily be checked usig the results of II above, they are ramified at ail primes dividig the order of the Galois group) Schur also foud A extesios of Q for odd uramified outside \ (see [S-2] ad [S-3]) This raises the questio, give a simple group G, does there exist a G extesio of Q uramified outside the order of G?

7 2 The origial proof of Schur utilized the followig resuit : Theorem (Schur 1929 [S-l]) Let 1 k<h be itegers The there exists a prime umber p>k which divides oe of the followig itegers : The proof uses Tschebyshev's method Schur eeded this resuit to demostrate the irreducibility of f, which we were able to obtai by elemetary meas However, Schur obtaied much more He : proved Theorem (Schur, 1929, [S-l]) Let a Q,a 19,a be itegers such that (a o,a 9 \) = 1 The is irreducible 3 Exercises (a) Calculate the Galois Groups of the Followig polyomials : (1) (Laguerre) (2) (3) (Hermite) (b) Calculate the discrimiats of polyomials (See [S-2] ad [S-3]) Usig either Schur's Criterio above or the methods of this ote, détermie the irreducibility of as may polyomials as you ca of thèse (c) For each prime p, détermie the iertia subgroups of G above p (Note, we hâve oly doe this whe is a power of p)

8 REFERENCES [Ha] Hall, M Theory of Group s Macmilla 1959 [H] Hilbert, D Über die Irreduzibilitât gaze ratioale Fuktioe mit gaz zahligekoeffiziete Jour, fur Math Bd 110 (1892), [H-W] Hardy, G H ad E M Wright A Itroductio to the Theory of Numbers, fifth éd, Clavedo Press, Oxford, 1979 [J-l] Jorda, C Traité des Substitutios et des Equatios Algébriques Gauthier- Villars, Paris (1870) [J-2] Sur la limite de trasivité des groupes o alterés Bul Soc Math Frace 1 ( ), [S-l] Schur, I Eiige Sàtze ùber Primzahle mit Aweduge auf Irreduzibilitàts fragei (1929) Gesammelte Abhadluge, Bad 111, No 64, [S-2] Gleichuge ohe Affekt (1930) Gesammelte Abhadluge, Bad 111, No 67, [S-3] Affektlose Gleichuge i der Théorie der Laguerresche ud Her miteschepolyôme (1931) Gesammelte Abhadluge, Bad 111, Nr 70, [T] Tschebyshev, P I Sur la totalité des ombres premiers iférieurs à ue limite doée J de Math 17 (1852), (Reçu le 22 août 1986) Robert F Colema Uiversity of Califoria Departmet of Mathematics Berkeley, Califoria USA Added i proof For some iterestig ew results^cocerig the Galois theory of the polyomials metioed above see: Â5Â W Feit, ad Â7Â 5 7 are Galois groups over umber fields<