Research Article On the Rank of Elliptic Curves in Elementary Cubic Extensions


 Delphia Small
 2 years ago
 Views:
Transcription
1 Numbers Volume 2015, Article ID , 4 pages Research Article On the Rank of Elliptic Curves in Elementary Cubic Extensions Rintaro Kozuma College of International Management, Ritsumeikan Asia Pacific University, Oita , Japan Correspondence should be addressed to Rintaro Kozuma; Received 21 July 2015; Accepted 6 September 2015 Academic Editor: Andrej Dujella Copyright 2015 Rintaro Kozuma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We give a method for explicitly constructing an elementary cubic extension L over which an elliptic curve E D :y 2 +Dy=x 3 (D Q ) has MordellWeil rank of at least a given positive integer by finding a close connection between a 3isogeny of E D and a generic polynomial for cyclic cubic extensions. In our method, the extension degree [L : Q] often becomes small. 1. Introduction Let E be an elliptic curve defined over a number field F.Itis wellknown that the MordellWeil group E(F) of Frational points on E forms a finitely generated abelian group, and its rank rank E(F) = dim Q E(F) Z Q is of great interest in arithmetic geometry. The present paper is motivated by the following general problem: To understand the behavior of ranks of elliptic curves in towers of finite extensions over F. This problem dates back at least to [1], which first introduced an Iwasawa theory for elliptic curves. In the context of Iwasawa theory, Kurcanov [2] showed that if an elliptic curve E over Q without complex multiplication has good reduction at a prime number p > 3 satisfying a mild condition then E has infinite rank in a certain Z p extension, and Harris [3] showed that if E is a modular elliptic curve over F having good ordinary reduction at p and a specific Frational point arisingfromthemodularcurvethenthereisapadic Lie extension of F in which the rank of E grows infinitely. On the other hand, Kida [4] constructed a tower of elementary 2extensions in which the rank of y 2 = x 3 n 2 x becomes arbitrarily large by using a result in theory of congruent numbers. Also, Dokchitser [5] proved that for any elliptic curve E over F there are infinitely many cubic extensions K/F so that the rank of E increases in K/F. In the present paper, we consider the specific curve E D :y 2 +Dy=x 3, D Q, (1) and give an explicit construction of an elementary cubic extension over which the elliptic curve E D is of rank of at least a given positive integer l by finding a close connection between a 3isogeny of E D and a generic polynomial for cyclic cubic extensions. We shall call an extension K/F an elementary cubic extension if K is a finite compositum of cubic extensions over F. Compared with related results as above, in our method, the extension degree [L : Q] often becomes small. The question of finding a small elementary cubic extension with rank l seems to be of independent interest. 2. Main Results Let S D ={b Z D=a 2 +3ab+9b 2 (a, b Z, b>0), where a is relatively prime to b}. We denote by F a fixed algebraic closure of F. LetF( 3 S D ) denote the field obtained by adjoining a real number 3 b for each b S D to F. SincethesetS D is finite, the extension F( 3 S D )/F is also finite. The main result of the paper is the following theorem. Theorem 1. Let D be any squarefree product of distinct primes p 1,...,p k Z satisfying p i 1(mod.Then,foranypositive (2)
2 2 Numbers Table 1: rank k. D L rank Q Q Q Q Q( 17) Q( Q( 61) Q( Q( 307) 3 3 integer l( k),wecanconstructasubfieldl of Q( 3 S D ) such that rank l, [L :Q] 3 l. ( As we will see in Section 4, the field L in Theorem 1 is effectively computable from the decomposition of the primes p 1,...,p k in Z[ζ 3 ],whereζ 3 denotes a primitive cube root of unity. See Remark 4 for this computation. It will also turn out in Section 4 that the field L is constructed using certain products of prime elements in Z[ζ 3 ] above each prime p i, and in particular L depends only on the primes p 1,...,p k ( 1(mod ) and a positive integer l( k). One question arises here: how small can [L : Q] be when p 1,...,p k vary through the set of all primes p 1(mod for a fixed integer l( k)? We give some examples in the case l=k(table 1). Corollary 2. Let D be any squarefree product of distinct primes p 1,...,p k Z satisfying p i 1(mod.Then,forany positive integer l,wecanconstructasubfieldl of Q( 3 S dd, 3 d) with some integer d such that rank E D (L ) l, [L : Q] 3 l+1. (4) Note that our construction in Theorem1 (resp., Corollary 2) differs from Dokchitser s [5], and the extension degree [L : Q] (resp., [L : Q])isoftensmallerthanorequal to 3 l 1 (resp., 3 l ). 3. A Connection between the Elliptic Curve E D and a Generic Polynomial for Cyclic Cubic Extensions In this section, let F be a perfect field with characteristic not equal to 3,andletD be any element in F.Theellipticcurve E D has a rational 3torsion point (0, 0) and admits a 3isogeny φ:e D E D =E D / (0, 0) over F.The3isogeny φ can be explicitly written as φ:e D E D (x, y) ( x3 +D 2 Dx 2, ( y/d)3 + 3 ( y/d) 1 ( y/d) ( y/d 1) ). (5) The isogenous elliptic curve is given by the form E D :t 2 +3t+9=Ds 3. (6) Taking Galois cohomology of the short exact sequence 0 φ E D [φ] E D E D 0yields a Kummer map E D (F) φ(e D (F)) H1 (F, E D [φ]) (7) P {σ φ 1 (P) σ 1 }. Combining this map with the map H 1 (F, E D [φ]) = Hom (Gal ( F, Z 3Z ) Gal ( C 3 ξ F Ker ξ, where Gal(C 3 /F) denotes the set of all cyclic cubic extensions over F together with F, Gal ( C 3 := { K F : a Galois extension in F Gal (K C 3 Z 3Z }, we have a map δ F : E D (F) φ(e D (F)) Gal (C 3 P F (φ 1 (P)). (8) (9) (10) It is wellknown that the set Gal(C 3 /F) is parametrized by a generic polynomial for C 3 extensions [6] g (x; t) =x 3 +tx 2 (t+ x+1 with t F, (11) which has discriminant (t 2 +3t+9) 2.WeshalldenotebyK t the minimal splitting field of g(x; t) over F 0 (t), wheref 0 stands for the prime field in F. Bythepropertyofg(x; t), forany K/F Gal(C 3 /F) there is some t Fso as to be K=K t F in F. The following proposition is the key observation in the present paper. Proposition 3. The connecting homomorphism δ F is given by δ F (s, t) = K t F for any (s, t) E D (F). Furthermore,inthe case where F is a number field, for any prime ideal p of F not dividing 3,thepcomponent of the conductor of K t F/F is p if ] p (Ds 3 )>0, ] p (D) 0(mod (12) 1 otherwise. Here ] p :F p Z denotes the normalized valuation of the padic completion F p.
3 Numbers 3 Proof. For any (s, t) E D (F), takeapoint(x, y) on E D satisfying φ(x, y) = (s, t). Bytheformoftheycoordinate of (5), the splitting field K t coincides with F 0 (t, y). Here F(x, y) = F(x) = F(y). Thelatterpartoftheproposition follows from the equation t 2 +3t+9 = Ds 3 by using Proposition 8.1 in [7]. 4. A Method for Constructing an Elementary Cubic Extension From now on, let D be a squarefree product of distinct primes p 1,...,p k Z satisfying p i 1(mod, and let l be a positive integer k. RecallaresultofKomatsu[8]thatthesetofallcycliccubic fields (i.e., cyclic cubic extensions over Q) of conductor D= p 1 p k has a onetoone correspondence to the subset of the algebraic torus T(Q) =Q { } consisting of 2 k 1 elements of the form c p1 ± T c p2 ± T ± T c pk 1 ± T c pk, (1 where + T is the composition of T(Q) given by a/b+ T c/d = (a/b c/d 1)/(a/b + c/d + 1), T denotes the inverse of + T given by T a/b = a/b 1, andc pi =e pi /f pi T(Q) is a rational number so that the pair (e pi,f pi ) of integers is unique in the sense of Lemma 2.1 in [8] (especially f pi 0(mod, whichwillbeusedinthefollowing)andsatisfies e 2 p i +e pi f pi +f 2 p i =(e pi ζ 3 )(e pi ζ 2 3 )=p i. (14) There is a group isomorphism φ : T G m mapping t to (t ζ 3 )/(t ζ 1 3 ),whereg m denotes the multiplicative group. Then the composition + T is written as a/b+ T c/d = φ 1 (φ(a/b)φ(c/d)). Now, let π i := e pi ζ 3, which is a prime element in Z[ζ 3 ] dividing p i,andletα denote the complex conjugation of α.foreachi(1 i l),let t i := { 3(c p1 + T + T c pk ) if i=1 { 3(c { p1 + T + T c pi 1 T c pi + T c pi+1 + T + T c pk ) if i =1. (15) Since f pi 0(mod for each i, there exist unique integers a i, b i so that a i 3b i ζ 3 = { π 1 π k if i=1 { π { 1 π i 1 π i π i+1 π k if i =1, (16) where a i is relatively prime to 3b i.thenφ(t i / = (a i 3b i ζ 3 )/(a i 3b i ζ 3 ),andthust i =3 φ 1 (φ(t i /) = a i /b i. Let L:=Q({ 3 b i 1 i l}). Remark 4. In order to calculate t i =a i /b i,wehaveonlyto know the values c pi by the definition of t i. For example, one can use Table 5.1 in [8], in which the values c p for p < 1000 arelisted.fromtheconstructionofl,thefieldl depends only on D=p 1 p k and a positive integer l( k). Lemma 5. Every prime number p i (1 i l)is unramified in L. Proof. Since p i does not divide 3b 1 b l and is unramified in Q(ζ 3 ),itturnsoutthatp i is unramified in Q(ζ 3,{ 3 b i 1 i l}) by Kummer theory. Hence, p i is also unramified in L=Q({ 3 b i 1 i l}). Proposition 6. K ti L Im δ L {L} for any i(1 i l). Proof. Since (a i 3b i ζ 3 )(a i 3b i ζ 3 )=D,wehave t 2 i +3t i +9=D( 3 b i 2 ) 3. (17) Thus ( b 3 2 i,ti ). ItfollowsfromProposition3that δ L ( b 3 2 i,ti ) = K ti L, and hence K ti L Im δ L.Combining Proposition 3 with Lemma 5 yields K ti L =L. Proposition 7. dim /φ() l. Proof. One can easily verify that (0, 3ζ 3 ) is a 3torsion point in E D (L(ζ 3 )), and the rational function f:=(t 3ζ 3 )/D 2 on E D has a zero of order 3 at (0, 3ζ 3 ) and a pole of order 3 at infinity and satisfies f φ=( y Dζ 3 3 Dx ). (18) Then there exists an injective homomorphism (see [9] for details): E D (L (ζ 3 )) φ(e D (L (ζ 3 ))) L(ζ L(ζ 3 ) 3 (s, t) D (t 3ζ 3 )L(ζ 3 ) 3 (0, 3ζ 3 ) D 2 L(ζ 3 ) 3. (19) Combining this with the natural map φ() E D (L (ζ 3 )) φ(e D (L (ζ 3 ))), (20) we have the homomorphism φ() L(ζ L(ζ 3 ) 3 (21) (s, t) D (t 3ζ 3 )L(ζ 3 ) 3. By Proposition 6, the point ( b 3 2 i,ti ) corresponds to D(t i 3ζ 3 )L(ζ 3 ) 3 = D(a i 3b i ζ 3 )L(ζ 3 ) 3 via the above homomorphism. It thus suffices to show that {D(a i 3b i ζ 3 ) 1 i l}is linearly independent in L(ζ 3 ) /L(ζ 3 ) 3.Thenthe points {( b 3 2 i,ti ) 1 i l}must be linearly independent in /φ(). Assume that γ:= {D (a i 3b i ζ 3 )} r i L(ζ 3 ) 3 (22) 1 i l
4 4 Numbers for some integers {r i } 1 i l.foreachi, letp i be a prime ideal of L(ζ 3 ) above the prime element π i Q(ζ 3 ).Forany prime ideal p of L(ζ 3 ), we denote by ] p : L(ζ 3 ) p Z the normalized (additive) valuation of the padic completion L(ζ 3 ) p.bylemma5,wehave] pi (π i )=1.Then l r { i 0(mod i=1 ] pj (γ) = l if j=1 { r i +r j 0(mod if j =1, { i=1 (2 and hence r i 0(mod for each i.thisprovesthat{d(a i 3b i ζ 3 ) 1 i l}is linearly independent in L(ζ 3 ) /L(ζ 3 ) 3. Let φ : E D E D be the dual isogeny to φ. Sincethe composition φ φ(resp., φ φ) is the multiplicationby3 map on E D (resp., E D ), we have an exact sequence of F 3 vector spaces 0 and thus [ φ] φ( [3]) 0, φ() rank = dim dim φ() φ [3] φ() + dim F 3 φ() [ φ] φ( [3]) dim [3]. (24) (25) Since ζ 3 L,thegroup[ φ] is trivial and the group [3] is generated by the point (0, 0).Weseethat rank = dim φ() + dim F 3 φ() 1. (26) ProofofCorollary2. By Theorem 1, we have only to consider the case l>k.(inthecasewherel k,taked=1.) Take distinct prime numbers {q 1,...,q l k } not dividing D with q i 1(mod. Letd:=q 1 q l k. Then, applying Theorem 1 to the elliptic curve E dd,thereexistssomesubfieldl Q( 3 S dd ) so as to be rank E dd (L) l with [L : Q] 3 l.letl := L( 3 d). Then, it is easily seen that E dd is isomorphic over L to E D. Therefore, rank E D (L )=rank E dd (L ) rank E dd (L) l. Here [L : Q] 3 [L:Q] 3 l+1. Conflict of Interests The author declares that there is no conflict of interests regarding the publication of this paper. Acknowledgment The author is supported by APU Academic Research Subsidy. References [1] B. Mazur, Rational points of abelian varieties with values in towers of number fields, Inventiones Mathematicae, vol. 18, pp , [2] P. F. Kurcanov, Elliptic curves of infinite rank over Γ extensions, Mathematics of the USSRSbornik, vol. 19, no. 2, pp , [3] M. Harris, Systematic growth of MordellWeil groups of abelian varieties in towers of number fields, Inventiones Mathematicae,vol.51,no.2,pp ,1979. [4] M. Kida, On the rank of an elliptic curve in elementary 2extensions, Japan Academy. Proceedings A. Mathematical Sciences,vol.69,no.10,pp ,1993. [5] T. Dokchitser, Ranks of elliptic curves in cubic extensions, Acta Arithmetica,vol.126,no.4,pp ,2007. [6] J.P. Serre, Topics in Galois Theory, Second Edition,vol.1of Research Notes in Mathematics,CRCPress,2007. [7] R. Kozuma, Elliptic curves related to cyclic cubic extensions, International Number Theory, vol.5,no.4,pp , [8] T. Komatsu, Cyclic cubic field with explicit Artin symbols, Tokyo Mathematics, vol. 30, no. 1, pp , [9]J.H.Silverman,The Arithmetic of Elliptic Curves, vol.106of Graduate Texts in Mathematics, Springer, ProofofTheorem1. By Proposition 7, it suffices to show that (0, 0) φ().then dim φ() 1, and hence rank l. (27) Assume that there exists some point Q such that φ(q) = (0, 0). ThenQ [3]. Itfollowsfromthe3 division polynomial 3x(x 3 27D 2 ) for E D that there is no nontrivial 3torsion point in. Therefore, Q must be the trivial element in [3]. This contradicts the assumption, and hence (0, 0) φ().
5 Operations Research Decision Sciences Applied Mathematics Algebra Probability and Statistics The Scientific World Journal International Differential Equations Submit your manuscripts at International Combinatorics Mathematical Physics Complex Analysis International Mathematics and Mathematical Sciences Mathematical Problems in Engineering Mathematics Discrete Mathematics Discrete Dynamics in Nature and Society Function Spaces Abstract and Applied Analysis International Stochastic Analysis Optimization
ON GALOIS REALIZATIONS OF THE 2COVERABLE SYMMETRIC AND ALTERNATING GROUPS
ON GALOIS REALIZATIONS OF THE 2COVERABLE SYMMETRIC AND ALTERNATING GROUPS DANIEL RABAYEV AND JACK SONN Abstract. Let f(x) be a monic polynomial in Z[x] with no rational roots but with roots in Q p for
More informationThe Dirichlet Unit Theorem
Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if
More informationAlex, I will take congruent numbers for one million dollars please
Alex, I will take congruent numbers for one million dollars please Jim L. Brown The Ohio State University Columbus, OH 4310 jimlb@math.ohiostate.edu One of the most alluring aspectives of number theory
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationGalois representations with open image
Galois representations with open image Ralph Greenberg University of Washington Seattle, Washington, USA May 7th, 2011 Introduction This talk will be about representations of the absolute Galois group
More informationFACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z
FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization
More informationI. GROUPS: BASIC DEFINITIONS AND EXAMPLES
I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called
More informationFactoring of Prime Ideals in Extensions
Chapter 4 Factoring of Prime Ideals in Extensions 4. Lifting of Prime Ideals Recall the basic AKLB setup: A is a Dedekind domain with fraction field K, L is a finite, separable extension of K of degree
More informationCyclotomic Extensions
Chapter 7 Cyclotomic Extensions A cyclotomic extension Q(ζ n ) of the rationals is formed by adjoining a primitive n th root of unity ζ n. In this chapter, we will find an integral basis and calculate
More information1. Recap: What can we do with Euler systems?
Euler Systems and Lfunctions These are notes for a talk I gave at the Warwick Number Theory study group on Euler Systems, June 2015. The results stated make more sense when taken in step with the rest
More information3 1. Note that all cubes solve it; therefore, there are no more
Math 13 Problem set 5 Artin 11.4.7 Factor the following polynomials into irreducible factors in Q[x]: (a) x 3 3x (b) x 3 3x + (c) x 9 6x 6 + 9x 3 3 Solution: The first two polynomials are cubics, so if
More informationMonogenic Fields and Power Bases Michael Decker 12/07/07
Monogenic Fields and Power Bases Michael Decker 12/07/07 1 Introduction Let K be a number field of degree k and O K its ring of integers Then considering O K as a Zmodule, the nicest possible case is
More informationIdeal Class Group and Units
Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals
More informationOn the generation of elliptic curves with 16 rational torsion points by Pythagorean triples
On the generation of elliptic curves with 16 rational torsion points by Pythagorean triples Brian Hilley Boston College MT695 Honors Seminar March 3, 2006 1 Introduction 1.1 Mazur s Theorem Let C be a
More informationField Fundamentals. Chapter 3. 3.1 Field Extensions. 3.1.1 Definitions. 3.1.2 Lemma
Chapter 3 Field Fundamentals 3.1 Field Extensions If F is a field and F [X] is the set of all polynomials over F, that is, polynomials with coefficients in F, we know that F [X] is a Euclidean domain,
More informationThe Chebotarev Density Theorem
The Chebotarev Density Theorem Hendrik Lenstra 1. Introduction Consider the polynomial f(x) = X 4 X 2 1 Z[X]. Suppose that we want to show that this polynomial is irreducible. We reduce f modulo a prime
More informationit is easy to see that α = a
21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UF. Therefore
More informationModule MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013
Module MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013 D. R. Wilkins Copyright c David R. Wilkins 1997 2013 Contents A Cyclotomic Polynomials 79 A.1 Minimum Polynomials of Roots of
More informationIntroduction to finite fields
Introduction to finite fields Topics in Finite Fields (Fall 2013) Rutgers University Swastik Kopparty Last modified: Monday 16 th September, 2013 Welcome to the course on finite fields! This is aimed at
More informationGROUPS ACTING ON A SET
GROUPS ACTING ON A SET MATH 435 SPRING 2012 NOTES FROM FEBRUARY 27TH, 2012 1. Left group actions Definition 1.1. Suppose that G is a group and S is a set. A left (group) action of G on S is a rule for
More informationEMBEDDING DEGREE OF HYPERELLIPTIC CURVES WITH COMPLEX MULTIPLICATION
EMBEDDING DEGREE OF HYPERELLIPTIC CURVES WITH COMPLEX MULTIPLICATION CHRISTIAN ROBENHAGEN RAVNSHØJ Abstract. Consider the Jacobian of a genus two curve defined over a finite field and with complex multiplication.
More informationGroup Theory. Contents
Group Theory Contents Chapter 1: Review... 2 Chapter 2: Permutation Groups and Group Actions... 3 Orbits and Transitivity... 6 Specific Actions The Right regular and coset actions... 8 The Conjugation
More informationCONTINUED FRACTIONS AND PELL S EQUATION. Contents 1. Continued Fractions 1 2. Solution to Pell s Equation 9 References 12
CONTINUED FRACTIONS AND PELL S EQUATION SEUNG HYUN YANG Abstract. In this REU paper, I will use some important characteristics of continued fractions to give the complete set of solutions to Pell s equation.
More informationQuadratic Equations in Finite Fields of Characteristic 2
Quadratic Equations in Finite Fields of Characteristic 2 Klaus Pommerening May 2000 english version February 2012 Quadratic equations over fields of characteristic 2 are solved by the well known quadratic
More informationBreaking The Code. Ryan Lowe. Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and
Breaking The Code Ryan Lowe Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and a minor in Applied Physics. As a sophomore, he took an independent study
More information11 Ideals. 11.1 Revisiting Z
11 Ideals The presentation here is somewhat different than the text. In particular, the sections do not match up. We have seen issues with the failure of unique factorization already, e.g., Z[ 5] = O Q(
More informationIRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction
IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationThe topology of smooth divisors and the arithmetic of abelian varieties
The topology of smooth divisors and the arithmetic of abelian varieties Burt Totaro We have three main results. First, we show that a smooth complex projective variety which contains three disjoint codimensionone
More informationThe fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992), 33 37) Bart de Smit
The fundamental group of the Hawaiian earring is not free Bart de Smit The fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992),
More informationIntroduction to Algebraic Geometry. Bézout s Theorem and Inflection Points
Introduction to Algebraic Geometry Bézout s Theorem and Inflection Points 1. The resultant. Let K be a field. Then the polynomial ring K[x] is a unique factorisation domain (UFD). Another example of a
More informationComputer Algebra for Computer Engineers
p.1/14 Computer Algebra for Computer Engineers Preliminaries Priyank Kalla Department of Electrical and Computer Engineering University of Utah, Salt Lake City p.2/14 Notation R: Real Numbers Q: Fractions
More informationA number field is a field of finite degree over Q. By the Primitive Element Theorem, any number
Number Fields Introduction A number field is a field of finite degree over Q. By the Primitive Element Theorem, any number field K = Q(α) for some α K. The minimal polynomial Let K be a number field and
More informationAlgebra 2. Rings and fields. Finite fields. A.M. Cohen, H. Cuypers, H. Sterk. Algebra Interactive
2 Rings and fields A.M. Cohen, H. Cuypers, H. Sterk A.M. Cohen, H. Cuypers, H. Sterk 2 September 25, 2006 1 / 20 For p a prime number and f an irreducible polynomial of degree n in (Z/pZ)[X ], the quotient
More informationMultiplicity. Chapter 6
Chapter 6 Multiplicity The fundamental theorem of algebra says that any polynomial of degree n 0 has exactly n roots in the complex numbers if we count with multiplicity. The zeros of a polynomial are
More informationALGORITHMS FOR ALGEBRAIC CURVES
ALGORITHMS FOR ALGEBRAIC CURVES SUMMARY OF LECTURE 4 1. SCHOOF S ALGORITHM Let K be a finite field with q elements. Let p be its characteristic. Let X be an elliptic curve over K. To simplify we assume
More informationTHE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS
THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear
More informationChapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.
Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize
More informationProof. The map. G n i. where d is the degree of D.
7. Divisors Definition 7.1. We say that a scheme X is regular in codimension one if every local ring of dimension one is regular, that is, the quotient m/m 2 is one dimensional, where m is the unique maximal
More information50. Splitting Fields. 50. Splitting Fields 165
50. Splitting Fields 165 1. We should note that Q(x) is an algebraic closure of Q(x). We know that is transcendental over Q. Therefore, p must be transcendental over Q, for if it were algebraic, then (
More informationminimal polyonomial Example
Minimal Polynomials Definition Let α be an element in GF(p e ). We call the monic polynomial of smallest degree which has coefficients in GF(p) and α as a root, the minimal polyonomial of α. Example: We
More informationNOTES ON LINEAR TRANSFORMATIONS
NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all
More informationCONTINUED FRACTIONS, PELL S EQUATION, AND TRANSCENDENTAL NUMBERS
CONTINUED FRACTIONS, PELL S EQUATION, AND TRANSCENDENTAL NUMBERS JEREMY BOOHER Continued fractions usually get shortchanged at PROMYS, but they are interesting in their own right and useful in other areas
More informationReal quadratic fields with class number divisible by 5 or 7
Real quadratic fields with class number divisible by 5 or 7 Dongho Byeon Department of Mathematics, Seoul National University, Seoul 151747, Korea Email: dhbyeon@math.snu.ac.kr Abstract. We shall show
More informationResearch Article The General Traveling Wave Solutions of the Fisher Equation with Degree Three
Advances in Mathematical Physics Volume 203, Article ID 65798, 5 pages http://dx.doi.org/0.55/203/65798 Research Article The General Traveling Wave Solutions of the Fisher Equation with Degree Three Wenjun
More informationFINITE FIELDS KEITH CONRAD
FINITE FIELDS KEITH CONRAD This handout discusses finite fields: how to construct them, properties of elements in a finite field, and relations between different finite fields. We write Z/(p) and F p interchangeably
More informationPart V, Abstract Algebra CS131 Mathematics for Computer Scientists II Note 29 RINGS AND FIELDS
CS131 Part V, Abstract Algebra CS131 Mathematics for Computer Scientists II Note 29 RINGS AND FIELDS We now look at some algebraic structures which have more than one binary operation. Rings and fields
More informationGeneric Polynomials of Degree Three
Generic Polynomials of Degree Three Benjamin C. Wallace April 2012 1 Introduction In the nineteenth century, the mathematician Évariste Galois discovered an elegant solution to the fundamental problem
More informationTwo classes of ternary codes and their weight distributions
Two classes of ternary codes and their weight distributions Cunsheng Ding, Torleiv Kløve, and Francesco Sica Abstract In this paper we describe two classes of ternary codes, determine their minimum weight
More informationr + s = i + j (q + t)n; 2 rs = ij (qj + ti)n + qtn.
Chapter 7 Introduction to finite fields This chapter provides an introduction to several kinds of abstract algebraic structures, particularly groups, fields, and polynomials. Our primary interest is in
More information10 Splitting Fields. 2. The splitting field for x 3 2 over Q is Q( 3 2,ω), where ω is a primitive third root of 1 in C. Thus, since ω = 1+ 3
10 Splitting Fields We have seen how to construct a field K F such that K contains a root α of a given (irreducible) polynomial p(x) F [x], namely K = F [x]/(p(x)). We can extendthe procedure to build
More informationA Minkowskistyle bound for the orders of the finite subgroups of the Cremona group of rank 2 over an arbitrary field.
A Minkowskistyle bound for the orders of the finite subgroups of the Cremona group of rank 2 over an arbitrary field JeanPierre Serre Let k be a field. Let Cr(k) be the Cremona group of rank 2 over k,
More informationOSTROWSKI FOR NUMBER FIELDS
OSTROWSKI FOR NUMBER FIELDS KEITH CONRAD Ostrowski classified the nontrivial absolute values on Q: up to equivalence, they are the usual (archimedean) absolute value and the padic absolute values for
More informationSOLVING POLYNOMIAL EQUATIONS BY RADICALS
SOLVING POLYNOMIAL EQUATIONS BY RADICALS Lee Si Ying 1 and Zhang DeQi 2 1 Raffles Girls School (Secondary), 20 Anderson Road, Singapore 259978 2 Department of Mathematics, National University of Singapore,
More informationMA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES
MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES 2016 47 4. Diophantine Equations A Diophantine Equation is simply an equation in one or more variables for which integer (or sometimes rational) solutions
More informationChapter 6 Finite sets and infinite sets. Copyright 2013, 2005, 2001 Pearson Education, Inc. Section 3.1, Slide 1
Chapter 6 Finite sets and infinite sets Copyright 013, 005, 001 Pearson Education, Inc. Section 3.1, Slide 1 Section 6. PROPERTIES OF THE NATURE NUMBERS 013 Pearson Education, Inc.1 Slide Recall that denotes
More informationFACTORING CERTAIN INFINITE ABELIAN GROUPS BY DISTORTED CYCLIC SUBSETS
International Electronic Journal of Algebra Volume 6 (2009) 95106 FACTORING CERTAIN INFINITE ABELIAN GROUPS BY DISTORTED CYCLIC SUBSETS Sándor Szabó Received: 11 November 2008; Revised: 13 March 2009
More information7. Some irreducible polynomials
7. Some irreducible polynomials 7.1 Irreducibles over a finite field 7.2 Worked examples Linear factors x α of a polynomial P (x) with coefficients in a field k correspond precisely to roots α k [1] of
More informationON HENSEL S ROOTS AND A FACTORIZATION FORMULA IN Z[[X]]
#A47 INTEGERS 4 (204) ON HENSEL S ROOTS AND A FACTORIZATION FORMULA IN Z[[X]] Daniel Birmaer Department of Mathematics, Nazareth College, Rochester, New Yor abirma6@naz.edu Juan B. Gil Penn State Altoona,
More informationPythagorean Triples, Complex Numbers, Abelian Groups and Prime Numbers
Pythagorean Triples, Complex Numbers, Abelian Groups and Prime Numbers Amnon Yekutieli Department of Mathematics Ben Gurion University email: amyekut@math.bgu.ac.il Notes available at http://www.math.bgu.ac.il/~amyekut/lectures
More informationINTRODUCTION TO MANIFOLDS IV. Appendix: algebraic language in Geometry
INTRODUCTION TO MANIFOLDS IV Appendix: algebraic language in Geometry 1. Algebras. Definition. A (commutative associatetive) algebra (over reals) is a linear space A over R, endowed with two operations,
More informationGalois Theory III. 3.1. Splitting fields.
Galois Theory III. 3.1. Splitting fields. We know how to construct a field extension L of a given field K where a given irreducible polynomial P (X) K[X] has a root. We need a field extension of K where
More informationCongruent Numbers, the Rank of Elliptic Curves and the Birch and SwinnertonDyer Conjecture. Brad Groff
Congruent Numbers, the Rank of Elliptic Curves and the Birch and SwinnertonDyer Conjecture Brad Groff Contents 1 Congruent Numbers... 1.1 Basic Facts............................... and Elliptic Curves.1
More informationOn the largest prime factor of x 2 1
On the largest prime factor of x 2 1 Florian Luca and Filip Najman Abstract In this paper, we find all integers x such that x 2 1 has only prime factors smaller than 100. This gives some interesting numerical
More informationMOP 2007 Black Group Integer Polynomials Yufei Zhao. Integer Polynomials. June 29, 2007 Yufei Zhao yufeiz@mit.edu
Integer Polynomials June 9, 007 Yufei Zhao yufeiz@mit.edu We will use Z[x] to denote the ring of polynomials with integer coefficients. We begin by summarizing some of the common approaches used in dealing
More informationADDITIVE GROUPS OF RINGS WITH IDENTITY
ADDITIVE GROUPS OF RINGS WITH IDENTITY SIMION BREAZ AND GRIGORE CĂLUGĂREANU Abstract. A ring with identity exists on a torsion Abelian group exactly when the group is bounded. The additive groups of torsionfree
More informationTHE CONGRUENT NUMBER PROBLEM
THE CONGRUENT NUMBER PROBLEM KEITH CONRAD 1. Introduction A right triangle is called rational when its legs and hypotenuse are all rational numbers. Examples of rational right triangles include Pythagorean
More informationCURVES WHOSE SECANT DEGREE IS ONE IN POSITIVE CHARACTERISTIC. 1. Introduction
Acta Math. Univ. Comenianae Vol. LXXXI, 1 (2012), pp. 71 77 71 CURVES WHOSE SECANT DEGREE IS ONE IN POSITIVE CHARACTERISTIC E. BALLICO Abstract. Here we study (in positive characteristic) integral curves
More informationIntroduction to Finite Fields (cont.)
Chapter 6 Introduction to Finite Fields (cont.) 6.1 Recall Theorem. Z m is a field m is a prime number. Theorem (Subfield Isomorphic to Z p ). Every finite field has the order of a power of a prime number
More informationThe van Hoeij Algorithm for Factoring Polynomials
The van Hoeij Algorithm for Factoring Polynomials Jürgen Klüners Abstract In this survey we report about a new algorithm for factoring polynomials due to Mark van Hoeij. The main idea is that the combinatorial
More informationCommutative Algebra seminar talk
Commutative Algebra seminar talk Reeve Garrett May 22, 2015 1 Ultrafilters Definition 1.1 Given an infinite set W, a collection U of subsets of W which does not contain the empty set and is closed under
More informationResearch Article Stability Analysis for HigherOrder Adjacent Derivative in Parametrized Vector Optimization
Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 510838, 15 pages doi:10.1155/2010/510838 Research Article Stability Analysis for HigherOrder Adjacent Derivative
More informationDiscrete Mathematics, Chapter 5: Induction and Recursion
Discrete Mathematics, Chapter 5: Induction and Recursion Richard Mayr University of Edinburgh, UK Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 5 1 / 20 Outline 1 Wellfounded
More informationCHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY
January 10, 2010 CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY The set of polynomials over a field F is a ring, whose structure shares with the ring of integers many characteristics.
More information4. Factor polynomials over complex numbers, describe geometrically, and apply to realworld situations. 5. Determine and apply relationships among syn
I The Real and Complex Number Systems 1. Identify subsets of complex numbers, and compare their structural characteristics. 2. Compare and contrast the properties of real numbers with the properties of
More informationNOTES ON CATEGORIES AND FUNCTORS
NOTES ON CATEGORIES AND FUNCTORS These notes collect basic definitions and facts about categories and functors that have been mentioned in the Homological Algebra course. For further reading about category
More informationFunctions and Equations
Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c
More informationG = G 0 > G 1 > > G k = {e}
Proposition 49. 1. A group G is nilpotent if and only if G appears as an element of its upper central series. 2. If G is nilpotent, then the upper central series and the lower central series have the same
More informationNOTES ON DEDEKIND RINGS
NOTES ON DEDEKIND RINGS J. P. MAY Contents 1. Fractional ideals 1 2. The definition of Dedekind rings and DVR s 2 3. Characterizations and properties of DVR s 3 4. Completions of discrete valuation rings
More informationChapter 13: Basic ring theory
Chapter 3: Basic ring theory Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 42, Spring 24 M. Macauley (Clemson) Chapter 3: Basic ring
More informationRevision of ring theory
CHAPTER 1 Revision of ring theory 1.1. Basic definitions and examples In this chapter we will revise and extend some of the results on rings that you have studied on previous courses. A ring is an algebraic
More informationsome algebra prelim solutions
some algebra prelim solutions David Morawski August 19, 2012 Problem (Spring 2008, #5). Show that f(x) = x p x + a is irreducible over F p whenever a F p is not zero. Proof. First, note that f(x) has no
More informationCHAPTER 5: MODULAR ARITHMETIC
CHAPTER 5: MODULAR ARITHMETIC LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN 1. Introduction In this chapter we will consider congruence modulo m, and explore the associated arithmetic called
More informationFinite Fields and ErrorCorrecting Codes
Lecture Notes in Mathematics Finite Fields and ErrorCorrecting Codes KarlGustav Andersson (Lund University) (version 1.01316 September 2015) Translated from Swedish by Sigmundur Gudmundsson Contents
More informationThe Ideal Class Group
Chapter 5 The Ideal Class Group We will use Minkowski theory, which belongs to the general area of geometry of numbers, to gain insight into the ideal class group of a number field. We have already mentioned
More informationQUADRATIC RECIPROCITY IN CHARACTERISTIC 2
QUADRATIC RECIPROCITY IN CHARACTERISTIC 2 KEITH CONRAD 1. Introduction Let F be a finite field. When F has odd characteristic, the quadratic reciprocity law in F[T ] (see [4, Section 3.2.2] or [5]) lets
More information8.1 Examples, definitions, and basic properties
8 De Rham cohomology Last updated: May 21, 211. 8.1 Examples, definitions, and basic properties A kform ω Ω k (M) is closed if dω =. It is exact if there is a (k 1)form σ Ω k 1 (M) such that dσ = ω.
More informationA REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE. 1. Introduction and Preliminaries
Acta Math. Univ. Comenianae Vol. LXVI, 2(1997), pp. 285 291 285 A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE E. T. BASKORO, M. MILLER and J. ŠIRÁŇ Abstract. It is well known that Moore digraphs do
More information6. Fields I. 1. Adjoining things
6. Fields I 6.1 Adjoining things 6.2 Fields of fractions, fields of rational functions 6.3 Characteristics, finite fields 6.4 Algebraic field extensions 6.5 Algebraic closures 1. Adjoining things The general
More informationCourse 421: Algebraic Topology Section 1: Topological Spaces
Course 421: Algebraic Topology Section 1: Topological Spaces David R. Wilkins Copyright c David R. Wilkins 1988 2008 Contents 1 Topological Spaces 1 1.1 Continuity and Topological Spaces...............
More informationSOME PROPERTIES OF FIBER PRODUCT PRESERVING BUNDLE FUNCTORS
SOME PROPERTIES OF FIBER PRODUCT PRESERVING BUNDLE FUNCTORS Ivan Kolář Abstract. Let F be a fiber product preserving bundle functor on the category FM m of the proper base order r. We deduce that the rth
More informationOn Bhargava s representations and Vinberg s invariant theory
On Bhargava s representations and Vinberg s invariant theory Benedict H. Gross Department of Mathematics, Harvard University Cambridge, MA 02138 gross@math.harvard.edu January, 2011 1 Introduction Manjul
More informationEXERCISES FOR THE COURSE MATH 570, FALL 2010
EXERCISES FOR THE COURSE MATH 570, FALL 2010 EYAL Z. GOREN (1) Let G be a group and H Z(G) a subgroup such that G/H is cyclic. Prove that G is abelian. Conclude that every group of order p 2 (p a prime
More informationPrime Numbers and Irreducible Polynomials
Prime Numbers and Irreducible Polynomials M. Ram Murty The similarity between prime numbers and irreducible polynomials has been a dominant theme in the development of number theory and algebraic geometry.
More informationELLIPTIC CURVES AND LENSTRA S FACTORIZATION ALGORITHM
ELLIPTIC CURVES AND LENSTRA S FACTORIZATION ALGORITHM DANIEL PARKER Abstract. This paper provides a foundation for understanding Lenstra s Elliptic Curve Algorithm for factoring large numbers. We give
More informationk, then n = p2α 1 1 pα k
Powers of Integers An integer n is a perfect square if n = m for some integer m. Taking into account the prime factorization, if m = p α 1 1 pα k k, then n = pα 1 1 p α k k. That is, n is a perfect square
More informationProblem Set 7  Fall 2008 Due Tuesday, Oct. 28 at 1:00
18.781 Problem Set 7  Fall 2008 Due Tuesday, Oct. 28 at 1:00 Throughout this assignment, f(x) always denotes a polynomial with integer coefficients. 1. (a) Show that e 32 (3) = 8, and write down a list
More informationElements of Abstract Group Theory
Chapter 2 Elements of Abstract Group Theory Mathematics is a game played according to certain simple rules with meaningless marks on paper. David Hilbert The importance of symmetry in physics, and for
More informationEvaluating large degree isogenies and applications to pairing based cryptography
Evaluating large degree isogenies and applications to pairing based cryptography Reinier Bröker, Denis Charles, and Kristin Lauter Microsoft Research, One Microsoft Way, Redmond, WA 98052, USA reinierb@microsoft.com,
More informationFirst and raw version 0.1 23. september 2013 klokken 13:45
The discriminant First and raw version 0.1 23. september 2013 klokken 13:45 One of the most significant invariant of an algebraic number field is the discriminant. One is tempted to say, apart from the
More information