Math 210B. Radical Towers and Roots of Unity 1. Motivation Let F ∈ Q[T] Be
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Solving Solvable Quintics
mathematics of computation volume 57,number 195 july 1991, pages 387-401 SOLVINGSOLVABLE QUINTICS D. S. DUMMIT Abstract. Let f{x) = x 5 +px 3 +qx 2 +rx + s be an irreducible polynomial of degree 5 with rational coefficients. An explicit resolvent sextic is constructed which has a rational root if and only if f(x) is solvable by radicals (i.e., when its Galois group is contained in the Frobenius group F20 of order 20 in the symmetric group S5). When f(x) is solvable by radicals, formulas for the roots are given in terms of p, q, r, s which produce the roots in a cyclic order. 1. Introduction It is well known that an irreducible quintic with coefficients in the rational numbers Q is solvable by radicals if and only if its Galois group is contained in the Frobenius group F20 of order 20, i.e., if and only if the Galois group is isomorphic to F20 , to the dihedral group DXQof order 10, or to the cyclic group Z/5Z. (More generally, for any prime p, it is easy to see that a solvable subgroup of the symmetric group S whose order is divisible by p is contained in the normalizer of a Sylow p-subgroup of S , cf. [1].) The purpose here is to give a criterion for the solvability of such a general quintic in terms of the existence of a rational root of an explicit associated resolvent sextic polynomial, and when this is the case, to give formulas for the roots analogous to Cardano's formulas for the general cubic and quartic polynomials (cf. -
Infinite Galois Theory
Infinite Galois Theory Haoran Liu May 1, 2016 1 Introduction For an finite Galois extension E/F, the fundamental theorem of Galois Theory establishes an one-to-one correspondence between the intermediate fields of E/F and the subgroups of Gal(E/F), the Galois group of the extension. With this correspondence, we can examine the the finite field extension by using the well-established group theory. Naturally, we wonder if this correspondence still holds if the Galois extension E/F is infinite. It is very tempting to assume the one-to-one correspondence still exists. Unfortu- nately, there is not necessary a correspondence between the intermediate fields of E/F and the subgroups of Gal(E/F)whenE/F is a infinite Galois extension. It will be illustrated in the following example. Example 1.1. Let F be Q,andE be the splitting field of a set of polynomials in the form of x2 p, where p is a prime number in Z+. Since each automorphism of E that fixes F − is determined by the square root of a prime, thusAut(E/F)isainfinitedimensionalvector space over F2. Since the number of homomorphisms from Aut(E/F)toF2 is uncountable, which means that there are uncountably many subgroups of Aut(E/F)withindex2.while the number of subfields of E that have degree 2 over F is countable, thus there is no bijection between the set of all subfields of E containing F and the set of all subgroups of Gal(E/F). Since a infinite Galois group Gal(E/F)normally have ”too much” subgroups, there is no subfield of E containing F can correspond to most of its subgroups. -
Casus Irreducibilis and Maple
48 Casus irreducibilis and Maple Rudolf V´yborn´y Abstract We give a proof that there is no formula which uses only addition, multiplication and extraction of real roots on the coefficients of an irreducible cubic equation with three real roots that would provide a solution. 1 Introduction The Cardano formulae for the roots of a cubic equation with real coefficients and three real roots give the solution in a rather complicated form involving complex numbers. Any effort to simplify it is doomed to failure; trying to get rid of complex numbers leads back to the original equation. For this reason, this case of a cubic is called casus irreducibilis: the irreducible case. The usual proof uses the Galois theory [3]. Here we give a fairly simple proof which perhaps is not quite elementary but should be accessible to undergraduates. It is well known that a convenient solution for a cubic with real roots is in terms of trigono- metric functions. In the last section we handle the irreducible case in Maple and obtain the trigonometric solution. 2 Prerequisites By N, Q, R and C we denote the natural numbers, the rationals, the reals and the com- plex numbers, respectively. If F is a field then F [X] denotes the ring of polynomials with coefficients in F . If F ⊂ C is a field, a ∈ C but a∈ / F then there exists a smallest field of complex numbers which contains both F and a, we denote it by F (a). Obviously it is the intersection of all fields which contain F as well as a. -
A Second Course in Algebraic Number Theory
A second course in Algebraic Number Theory Vlad Dockchitser Prerequisites: • Galois Theory • Representation Theory Overview: ∗ 1. Number Fields (Review, K; OK ; O ; ClK ; etc) 2. Decomposition of primes (how primes behave in eld extensions and what does Galois's do) 3. L-series (Dirichlet's Theorem on primes in arithmetic progression, Artin L-functions, Cheboterev's density theorem) 1 Number Fields 1.1 Rings of integers Denition 1.1. A number eld is a nite extension of Q Denition 1.2. An algebraic integer α is an algebraic number that satises a monic polynomial with integer coecients Denition 1.3. Let K be a number eld. It's ring of integer OK consists of the elements of K which are algebraic integers Proposition 1.4. 1. OK is a (Noetherian) Ring 2. , i.e., ∼ [K:Q] as an abelian group rkZ OK = [K : Q] OK = Z 3. Each can be written as with and α 2 K α = β=n β 2 OK n 2 Z Example. K OK Q Z ( p p [ a] a ≡ 2; 3 mod 4 ( , square free) Z p Q( a) a 2 Z n f0; 1g a 1+ a Z[ 2 ] a ≡ 1 mod 4 where is a primitive th root of unity Q(ζn) ζn n Z[ζn] Proposition 1.5. 1. OK is the maximal subring of K which is nitely generated as an abelian group 2. O`K is integrally closed - if f 2 OK [x] is monic and f(α) = 0 for some α 2 K, then α 2 OK . Example (Of Factorisation). -
Algebraic Number Theory
Algebraic Number Theory William B. Hart Warwick Mathematics Institute Abstract. We give a short introduction to algebraic number theory. Algebraic number theory is the study of extension fields Q(α1; α2; : : : ; αn) of the rational numbers, known as algebraic number fields (sometimes number fields for short), in which each of the adjoined complex numbers αi is algebraic, i.e. the root of a polynomial with rational coefficients. Throughout this set of notes we use the notation Z[α1; α2; : : : ; αn] to denote the ring generated by the values αi. It is the smallest ring containing the integers Z and each of the αi. It can be described as the ring of all polynomial expressions in the αi with integer coefficients, i.e. the ring of all expressions built up from elements of Z and the complex numbers αi by finitely many applications of the arithmetic operations of addition and multiplication. The notation Q(α1; α2; : : : ; αn) denotes the field of all quotients of elements of Z[α1; α2; : : : ; αn] with nonzero denominator, i.e. the field of rational functions in the αi, with rational coefficients. It is the smallest field containing the rational numbers Q and all of the αi. It can be thought of as the field of all expressions built up from elements of Z and the numbers αi by finitely many applications of the arithmetic operations of addition, multiplication and division (excepting of course, divide by zero). 1 Algebraic numbers and integers A number α 2 C is called algebraic if it is the root of a monic polynomial n n−1 n−2 f(x) = x + an−1x + an−2x + ::: + a1x + a0 = 0 with rational coefficients ai. -
GALOIS THEORY for ARBITRARY FIELD EXTENSIONS Contents 1
GALOIS THEORY FOR ARBITRARY FIELD EXTENSIONS PETE L. CLARK Contents 1. Introduction 1 1.1. Kaplansky's Galois Connection and Correspondence 1 1.2. Three flavors of Galois extensions 2 1.3. Galois theory for algebraic extensions 3 1.4. Transcendental Extensions 3 2. Galois Connections 4 2.1. The basic formalism 4 2.2. Lattice Properties 5 2.3. Examples 6 2.4. Galois Connections Decorticated (Relations) 8 2.5. Indexed Galois Connections 9 3. Galois Theory of Group Actions 11 3.1. Basic Setup 11 3.2. Normality and Stability 11 3.3. The J -topology and the K-topology 12 4. Return to the Galois Correspondence for Field Extensions 15 4.1. The Artinian Perspective 15 4.2. The Index Calculus 17 4.3. Normality and Stability:::and Normality 18 4.4. Finite Galois Extensions 18 4.5. Algebraic Galois Extensions 19 4.6. The J -topology 22 4.7. The K-topology 22 4.8. When K is algebraically closed 22 5. Three Flavors Revisited 24 5.1. Galois Extensions 24 5.2. Dedekind Extensions 26 5.3. Perfectly Galois Extensions 27 6. Notes 28 References 29 Abstract. 1. Introduction 1.1. Kaplansky's Galois Connection and Correspondence. For an arbitrary field extension K=F , define L = L(K=F ) to be the lattice of 1 2 PETE L. CLARK subextensions L of K=F and H = H(K=F ) to be the lattice of all subgroups H of G = Aut(K=F ). Then we have Φ: L!H;L 7! Aut(K=L) and Ψ: H!F;H 7! KH : For L 2 L, we write c(L) := Ψ(Φ(L)) = KAut(K=L): One immediately verifies: L ⊂ L0 =) c(L) ⊂ c(L0);L ⊂ c(L); c(c(L)) = c(L); these properties assert that L 7! c(L) is a closure operator on the lattice L in the sense of order theory. -
Pseudo Real Closed Field, Pseudo P-Adically Closed Fields and NTP2
Pseudo real closed fields, pseudo p-adically closed fields and NTP2 Samaria Montenegro∗ Université Paris Diderot-Paris 7† Abstract The main result of this paper is a positive answer to the Conjecture 5.1 of [15] by A. Chernikov, I. Kaplan and P. Simon: If M is a PRC field, then T h(M) is NTP2 if and only if M is bounded. In the case of PpC fields, we prove that if M is a bounded PpC field, then T h(M) is NTP2. We also generalize this result to obtain that, if M is a bounded PRC or PpC field with exactly n orders or p-adic valuations respectively, then T h(M) is strong of burden n. This also allows us to explicitly compute the burden of types, and to describe forking. Keywords: Model theory, ordered fields, p-adic valuation, real closed fields, p-adically closed fields, PRC, PpC, NIP, NTP2. Mathematics Subject Classification: Primary 03C45, 03C60; Secondary 03C64, 12L12. Contents 1 Introduction 2 2 Preliminaries on pseudo real closed fields 4 2.1 Orderedfields .................................... 5 2.2 Pseudorealclosedfields . .. .. .... 5 2.3 The theory of PRC fields with n orderings ..................... 6 arXiv:1411.7654v2 [math.LO] 27 Sep 2016 3 Bounded pseudo real closed fields 7 3.1 Density theorem for PRC bounded fields . ...... 8 3.1.1 Density theorem for one variable definable sets . ......... 9 3.1.2 Density theorem for several variable definable sets. ........... 12 3.2 Amalgamation theorems for PRC bounded fields . ........ 14 ∗[email protected]; present address: Universidad de los Andes †Partially supported by ValCoMo (ANR-13-BS01-0006) and the Universidad de Costa Rica. -
Number Theory and Graph Theory Chapter 3 Arithmetic Functions And
1 Number Theory and Graph Theory Chapter 3 Arithmetic functions and roots of unity By A. Satyanarayana Reddy Department of Mathematics Shiv Nadar University Uttar Pradesh, India E-mail: [email protected] 2 Module-4: nth roots of unity Objectives • Properties of nth roots of unity and primitive nth roots of unity. • Properties of Cyclotomic polynomials. Definition 1. Let n 2 N. Then, a complex number z is called 1. an nth root of unity if it satisfies the equation xn = 1, i.e., zn = 1. 2. a primitive nth root of unity if n is the smallest positive integer for which zn = 1. That is, zn = 1 but zk 6= 1 for any k;1 ≤ k ≤ n − 1. 2pi zn = exp( n ) is a primitive n-th root of unity. k • Note that zn , for 0 ≤ k ≤ n − 1, are the n distinct n-th roots of unity. • The nth roots of unity are located on the unit circle of the complex plane, and in that plane they form the vertices of an n-sided regular polygon with one vertex at (1;0) and centered at the origin. The following points are collected from the article Cyclotomy and cyclotomic polynomials by B.Sury, Resonance, 1999. 1. Cyclotomy - literally circle-cutting - was a puzzle begun more than 2000 years ago by the Greek geometers. In their pastime, they used two implements - a ruler to draw straight lines and a compass to draw circles. 2. The problem of cyclotomy was to divide the circumference of a circle into n equal parts using only these two implements. -
Monic Polynomials in $ Z [X] $ with Roots in the Unit Disc
MONIC POLYNOMIALS IN Z[x] WITH ROOTS IN THE UNIT DISC Pantelis A. Damianou A Theorem of Kronecker This note is motivated by an old result of Kronecker on monic polynomials with integer coefficients having all their roots in the unit disc. We call such polynomials Kronecker polynomials for short. Let k(n) denote the number of Kronecker polynomials of degree n. We describe a canonical form for such polynomials and use it to determine the sequence k(n), for small values of n. The first step is to show that the number of Kronecker polynomials of degree n is finite. This fact is included in the following theorem due to Kronecker [6]. See also [5] for a more accessible proof. The theorem actually gives more: the non-zero roots of such polynomials are on the boundary of the unit disc. We use this fact later on to show that these polynomials are essentially products of cyclotomic polynomials. Theorem 1 Let λ =06 be a root of a monic polynomial f(z) with integer coefficients. If all the roots of f(z) are in the unit disc {z ∈ C ||z| ≤ 1}, then |λ| =1. Proof Let n = degf. The set of all monic polynomials of degree n with integer coefficients having all their roots in the unit disc is finite. To see this, we write n n n 1 n 2 z + an 1z − + an 2z − + ··· + a0 = (z − zj) , − − j Y=1 where aj ∈ Z and zj are the roots of the polynomial. Using the fact that |zj| ≤ 1 we have: n |an 1| = |z1 + z2 + ··· + zn| ≤ n = − 1 n |an 2| = | zjzk| ≤ − j,k 2! . -
Galois Theory on the Line in Nonzero Characteristic
BULLETIN (New Series) OF THE AMERICANMATHEMATICAL SOCIETY Volume 27, Number I, July 1992 GALOIS THEORY ON THE LINE IN NONZERO CHARACTERISTIC SHREERAM S. ABHYANKAR Dedicated to Walter Feit, J-P. Serre, and e-mail 1. What is Galois theory? Originally, the equation Y2 + 1 = 0 had no solution. Then the two solutions i and —i were created. But there is absolutely no way to tell who is / and who is —i.' That is Galois Theory. Thus, Galois Theory tells you how far we cannot distinguish between the roots of an equation. This is codified in the Galois Group. 2. Galois groups More precisely, consider an equation Yn + axY"-x +... + an=0 and let ai , ... , a„ be its roots, which are assumed to be distinct. By definition, the Galois Group G of this equation consists of those permutations of the roots which preserve all relations between them. Equivalently, G is the set of all those permutations a of the symbols {1,2,...,«} such that (t>(aa^, ... , a.a(n)) = 0 for every «-variable polynomial (j>for which (j>(a\, ... , an) — 0. The co- efficients of <f>are supposed to be in a field K which contains the coefficients a\, ... ,an of the given polynomial f = f(Y) = Y" + alY"-l+--- + an. We call G the Galois Group of / over K and denote it by Galy(/, K) or Gal(/, K). This is Galois' original concrete definition. According to the modern abstract definition, the Galois Group of a normal extension L of a field K is defined to be the group of all AT-automorphisms of L and is denoted by Gal(L, K). -
Finite Fields: Further Properties
Chapter 4 Finite fields: further properties 8 Roots of unity in finite fields In this section, we will generalize the concept of roots of unity (well-known for complex numbers) to the finite field setting, by considering the splitting field of the polynomial xn − 1. This has links with irreducible polynomials, and provides an effective way of obtaining primitive elements and hence representing finite fields. Definition 8.1 Let n ∈ N. The splitting field of xn − 1 over a field K is called the nth cyclotomic field over K and denoted by K(n). The roots of xn − 1 in K(n) are called the nth roots of unity over K and the set of all these roots is denoted by E(n). The following result, concerning the properties of E(n), holds for an arbitrary (not just a finite!) field K. Theorem 8.2 Let n ∈ N and K a field of characteristic p (where p may take the value 0 in this theorem). Then (i) If p ∤ n, then E(n) is a cyclic group of order n with respect to multiplication in K(n). (ii) If p | n, write n = mpe with positive integers m and e and p ∤ m. Then K(n) = K(m), E(n) = E(m) and the roots of xn − 1 are the m elements of E(m), each occurring with multiplicity pe. Proof. (i) The n = 1 case is trivial. For n ≥ 2, observe that xn − 1 and its derivative nxn−1 have no common roots; thus xn −1 cannot have multiple roots and hence E(n) has n elements. -
Solvability by Radicals Rodney Coleman, Laurent Zwald
Solvability by radicals Rodney Coleman, Laurent Zwald To cite this version: Rodney Coleman, Laurent Zwald. Solvability by radicals. 2020. hal-03066602 HAL Id: hal-03066602 https://hal.archives-ouvertes.fr/hal-03066602 Preprint submitted on 15 Dec 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Solvability by radicals Rodney Coleman, Laurent Zwald December 10, 2020 Abstract In this note we present one of the fundamental theorems of algebra, namely Galois’s theorem concerning the solution of polynomial equations. We will begin with a study of cyclic extensions, before moving onto radical extensions. 1 Cyclic extensions We say that a finite extension E of a field F is cyclic if the Galois group Gal(E=F ) is cyclic. In this section we aim to present some elementary properties of such extensions. We begin with a preliminary result known as Hilbert’s theorem 90. EXTSOLVcycth1 Theorem 1 Let E=F be a finite cyclic Galois extension of degree n. We suppose that σ is a generator of the Galois group Gal(E=F ). If α 2 E, then NE=F (α) = 1 if and only if there exists ∗ β β 2 E such that α = σ(β) .