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Radical Extensions and Galois Groups

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Radical extensions and Galois groups een wetenschappelijke proeve op het gebied van de Natuurwetenschappen, Wiskunde en Informatica Proefschrift ter verkrijging van de graad van doctor aan de Radboud Universiteit Nijmegen op gezag van de Rector Magni¯cus prof. dr. C.W.P.M. Blom, volgens besluit van het College van Decanen in het openbaar te verdedigen op dinsdag 17 mei 2005 des namiddags om 3.30 uur precies door Mascha Honsbeek geboren op 25 oktober 1971 te Haarlem Promotor: prof. dr. F.J. Keune Copromotores: dr. W. Bosma dr. B. de Smit (Universiteit Leiden) Manuscriptcommissie: prof. dr. F. Beukers (Universiteit Utrecht) prof. dr. A.M. Cohen (Technische Universiteit Eindhoven) prof. dr. H.W. Lenstra (Universiteit Leiden) Dankwoord Zoals iedereen aan mijn cv kan zien, is het schrijven van dit proefschrift niet helemaal volgens schema gegaan. De eerste drie jaar wilde het niet echt lukken om resultaten te krijgen. Toch heb ik in die tijd heel ¯jn samengewerkt met Frans Keune en Wieb Bosma. Al klopte ik drie keer per dag aan de deur, steeds maakten ze de tijd voor mij. Frans en Wieb, dank je voor jullie vriendschap en begeleiding. Op het moment dat ik dacht dat ik het schrijven van een proefschrift misschien maar op moest geven, kwam ik in contact met Bart de Smit. Hij keek net een beetje anders tegen de wiskunde waarmee ik bezig was aan en hielp me daarmee weer op weg. Samen met Hendrik Lenstra kwam hij met het voorstel voor het onderzoek dat ik heb uitgewerkt in het eerste hoofdstuk, de kern van dit proefschrift. Daarnaast heeft Bart mij ook vanaf dat moment als copromotor begeleid. Bart, Hendrik, dank je wel voor jullie steun en goede ideeÄen. Daarnaast wil ik ook graag mijn leescommissie bedanken. Hendrik, Arjeh en Frits, dank je wel voor jullie inzet en goede suggesties. Ook vanuit mijn vrienden en ouders heb ik veel steun ontvangen. Voorzichtig in- formeerden ze soms of het eind al in zicht was. Speciaal wil ik Arnoud, Kirsten en Timo noemen. Pap, mam en al die anderen die er met mij in zijn blijven geloven, dank je wel. Verder ontwierp Jan een schitterende kaft voor dit proefschrift. Helaas kreeg ik geen toestemming om deze Nijntje-afbeelding te gebruiken. Tot slot, Richard, wil ik jou bedanken voor alle steun en hulp. Zonder jou had het proefschrift er niet zo mooi uitgezien. Mascha III IV Contents Dankwoord III Table of Contents V Introduction VII 1 Maximal radical extensions 1 1.1 Preliminaries . 2 1.2 Kummer theory . 5 1.3 Maximal Kummer extensions . 9 1.4 Separating the roots of unity . 11 1.5 Galois action on radicals . 15 1.6 The maximal Kummer subextension in L . 19 1.7 Determining the Galois group . 21 1.8 Computing the ¯eld D . 27 2 Sub¯elds of radical extensions 33 2.1 Known results . 34 2.2 First implication . 37 2.3 The Galois group . 41 2.4 Subgroups . 46 2.5 Conclusion of the proof . 48 3 Ramanujan's nested radicals 57 3.1 The problem and its history . 58 3.2 Denesting condition . 59 3.3 A special polynomial . 62 3.4 Comparison to Ramanujan's method . 65 4 Nested radicals of depth one 69 4.1 Counterexample to Zippel's conjecture . 70 4.2 Denesting without roots of unity . 71 V VI CONTENTS 4.3 Denesting allowing roots of unity . 73 5 Rogers-Ramanujan continued fraction 77 5.1 Introduction . 79 5.2 The modular function ¯eld of level 5 . 80 5.3 Galois theory . 85 5.4 The ray class ¯eld H5 . 90 5.5 Nested radicals . 94 Bibliography 97 Nederlandse samenvatting 101 Curriculum Vitae 105 Introduction The use of symbolic computing is one of the characteristics of a computer algebra package. For example, the number p2 is represented as a symbol with the property that its square is 2. This enables us to do exact calculations. Compared to numerical calculations, there also are some drawbacks concerning computations with radical expressions. For example, algebraically, one will not be able to distinguish between the positive and the negative square root of 2. If one wants to do a computation for one speci¯c choice of these roots sometimes a numerical estimate is necessary. Moreover, we often get complicated expressions although there is a simpler expression for the number that we are interested in. The simpli¯cation of radical expressions was an important motivation for the research in this thesis. Look at 3 3 ® = p5 + 2 p5 2; ¡ ¡ where, as in the rest of this introqduction, whenq we take a root of a positive real number, we mean the real positive root of this number. A numerical estimate of ® is given by 3 3 p5 + 2 p5 2 = 1:618033988 : : : 0:618033988 : : : : ¡ ¡ ¡ q q We see that ® is approximately equal to 1. Using symbolic computations we prove that ® actually equals 1. One can check that ® is a root of the polynomial x3 + 3x 4 = (x 1)(x2 + x + 4), which has only a single real root. Therefore ® has to ¡ ¡ equal 1. Intuitively the notation 1 is simpler than 3 p5 + 2 3 p5 2. To make this precise we introduce the notion of radicals and nesting depth.¡ ¡ p p Let K be a ¯eld of characteristic 0 and let K¹ be some ¯xed algebraic closure of K. We say that ® K¹ is a radical over K if there exists some n Z>0 with ®n K. We denote by2K(0) the ¯eld K itself and inductively de¯ne the2¯elds K (k) for 2k 1 as ¸ (k) (k 1) n (k 1) K = K ¡ ( ® K¹ with ® K ¡ for some n Z ): f 2 2 2 >0g We say that ® K¹ is a nested radical over K if there exists some k Z>0 with (k) 2 (k) (k 21) ® K . We say that ® has nesting depth k over K if ® K K ¡ . 2 2 n VII VIII It is desirable for the computer to produce the simplest possible expression for the outcome of a computation with nested radicals. Richard Zippel [40] gave the following identity 6 7p3 20 19 = 3 5=3 3 2=3: (1) ¡ ¡ q Following [5], [20], [21], [28], [40] we say thatp the righp t hand side of this equation is a denesting of the left hand side; the equality shows that some element that is clearly contained in K(2) is even contained in K(1). The element ® K¹ is a nested radical if and only if there exists a Galois extension L=K with ® L and2 a chain of ¯eld extensions 2 K = K K : : : K = L; 0 ½ 1 ½ ½ t such that for 1 i t the ¯eld Ki is generated by radicals over Ki 1. For example, · · ¡ if we take the nested radical ® = 6 7p3 20 19 from equality (1) over Q, then an obvious choice for the ¯rst ¯elds in this chain¡ would be p 6 6 Q Q p3 20 Q p3 20; 7p3 20 19 = Q 7p3 20 19 : ½ ½ ¡ ¡ ³ ´ µ q ¶ µq ¶ In general we have to adjoin all conjugates of ® to obtain a Galois extension. In this case however, it turns out that Q 6 7p3 20 19; ³ is Galois over Q. One can ¡ 6 check this using a computer algebra pac³pkage or by using´equality (2) below. If we adjoin su±ciently many roots of unity in the ¯rst step of the chain, then we see that Ki=Ki 1 is abelian for all i 1. In our example: ¡ ¸ 6 Q Q(³ ) Q p3 20; ³ Q 7p3 20 19; ³ : ½ 6 ½ 6 ½ ¡ 6 ³ ´ µq ¶ As K is a ¯eld of characteristic 0, an element ® is a nested radical if and only if the Galois group of the normal closure of K(®) over K is solvable ([22] Chapter VI). Given a chain of ¯eld extensions K = K0 K1 : : : Ks; for which Ki=Ki 1, for ½ ½ ½ ¡ i Z>0, is generated by a radical, and an element ® Ks of nesting depth t s Susan2 Landau [20] provides an algorithm that computes2 a symbolic representation· for ® in K(d), where d is the length of the derived series of the Galois closure of Ks=K. We have t d t + 1. Carl Cotner [7]·impro· ved this result. He showed that the nesting depth of ® is computable and gave an algorithm to ¯nd a symbolic representation of minimal nesting depth. For speci¯c types of nested radicals it is possible to give a simple algorithm to compute a symbolic representation of minimal depth. This we will do in chapters 3 and 4 for certain elements of K(2). Given a sub¯eld L of K(1), an element ± of L, and n (1) some n Z 2 we will prove that p± K implies that ± satis¯es a strong condition 2 ¸ 2 INTRODUCTION IX which may be phrased as `± is almost an n-th power in L'. Indeed, equality (1) holds because `7p3 20 19 is almost a sixth power in Q(p3 20 )'; we have ¡ 1 6 7p3 20 19 = p3 20 2 (2) ¡ 144 ¡ and ³ ´ 1 6 p3 20 2 = 3 1=12 p3 20 p3 8 = 3 5=3 3 2=3: r144 ¡ ¡ ¡ To prove our claim³ in general´ wpe use Galois³ groups´ ofpradical extensions.p If the ground ¯eld K contains the appropriate roots of unity and the extension is ¯nite then we have a Kummer extension and the Galois group is well known. We discuss Kummer theory in chapter 1. Moreover, we discuss non-¯nite Kummer extensions and describe the Galois group for certain radical extensions for which the ground ¯eld K does not contain the appropriate roots of unity.
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