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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. -
THE INTERSECTION of NORM GROUPS(I) by JAMES AX
THE INTERSECTION OF NORM GROUPS(i) BY JAMES AX 1. Introduction. Let A be a global field (either a finite extension of Q or a field of algebraic functions in one variable over a finite field) or a local field (a local completion of a global field). Let C(A,n) (respectively: A(A, n), JV(A,n) and £(A, n)) be the set of Xe A such that X is the norm of every cyclic (respectively: abelian, normal and arbitrary) extension of A of degree n. We show that (*) C(A, n) = A(A,n) = JV(A,n) = £(A, n) = A" is "almost" true for any global or local field and any natural number n. For example, we prove (*) if A is a number field and %)(n or if A is a function field and n is arbitrary. In the case when (*) is false we are still able to determine C(A, n) precisely. It then turns out that there is a specified X0e A such that C(A,n) = r0/2An u A". Since we always have C(A,n) =>A(A,n) =>N(A,n) r> £(A,n) =>A", there are thus two possibilities for each of the three middle sets. Determining which is true seems to be a delicate question; our results on this problem, which are incomplete, are presented in §5. 2. Preliminaries. We consider an algebraic number field A as a subfield of the field of all complex numbers. If p is a nonarchimedean prime of A then there is a natural injection A -> Ap where Ap denotes the completion of A at p. -
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. -
Solving Cubic Polynomials
Solving Cubic Polynomials 1.1 The general solution to the quadratic equation There are four steps to finding the zeroes of a quadratic polynomial. 1. First divide by the leading term, making the polynomial monic. a 2. Then, given x2 + a x + a , substitute x = y − 1 to obtain an equation without the linear term. 1 0 2 (This is the \depressed" equation.) 3. Solve then for y as a square root. (Remember to use both signs of the square root.) a 4. Once this is done, recover x using the fact that x = y − 1 . 2 For example, let's solve 2x2 + 7x − 15 = 0: First, we divide both sides by 2 to create an equation with leading term equal to one: 7 15 x2 + x − = 0: 2 2 a 7 Then replace x by x = y − 1 = y − to obtain: 2 4 169 y2 = 16 Solve for y: 13 13 y = or − 4 4 Then, solving back for x, we have 3 x = or − 5: 2 This method is equivalent to \completing the square" and is the steps taken in developing the much- memorized quadratic formula. For example, if the original equation is our \high school quadratic" ax2 + bx + c = 0 then the first step creates the equation b c x2 + x + = 0: a a b We then write x = y − and obtain, after simplifying, 2a b2 − 4ac y2 − = 0 4a2 so that p b2 − 4ac y = ± 2a and so p b b2 − 4ac x = − ± : 2a 2a 1 The solutions to this quadratic depend heavily on the value of b2 − 4ac. -
The Kronecker-Weber Theorem
The Kronecker-Weber Theorem Lucas Culler Introduction The Kronecker-Weber theorem is one of the earliest known results in class field theory. It says: Theorem. (Kronecker-Weber-Hilbert) Every abelian extension of the rational numbers Q is con- tained in a cyclotomic extension. Recall that an abelian extension is a finite field extension K/Q such that the galois group Gal(K/Q) th is abelian, and a cyclotomic extension is an extension of the form Q(ζ), where ζ is an n root of unity. This paper consists of two proofs of the Kronecker-Weber theorem. The first is rather involved, but elementary, and uses the theory of higher ramification groups. The second is a simple application of the main results of class field theory, which classifies abelian extension of an arbitrary number field. An Elementary Proof Now we will present an elementary proof of the Kronecker-Weber theoerem, in the spirit of Hilbert’s original proof. The particular strategy used here is given as a series of exercises in Marcus [1]. Minkowski’s Theorem We first prove a classical result due to Minkowski. Theorem. (Minkowski) Any finite extension of Q has nonzero discriminant. In particular, such an extension is ramified at some prime p ∈ Z. Proof. Let K/Q be a finite extension of degree n, and let A = OK be its ring of integers. Consider the embedding: r s A −→ R ⊕ C x 7→ (σ1(x), ..., σr(x), τ1(x), ..., τs(x)) where the σi are the real embeddings of K and the τi are the complex embeddings, with one embedding chosen from each conjugate pair, so that n = r + 2s. -
F[X] Be an Irreducible Cubic X 3 + Ax 2 + Bx + Cw
Math 404 Assignment 3. Due Friday, May 3, 2013. Cubic equations. Let f(x) F [x] be an irreducible cubic x3 + ax2 + bx + c with roots ∈ x1, x2, x3, and splitting field K/F . Since each element of the Galois group permutes the roots, G(K/F ) is a subgroup of S3, the group of permutations of the three roots, and [K : F ] = G(K/F ) divides (S3) = 6. Since [F (x1): F ] = 3, we see that [K : F ] equals ◦ ◦ 3 or 6, and G(K/F ) is either A3 or S3. If G(K/F ) = S3, then K has a unique subfield of dimension 2, namely, KA3 . We have seen that the determinant J of the Jacobian matrix of the partial derivatives of the system a = (x1 + x2 + x3) − b = x1x2 + x2x3 + x3x1 c = (x1x2x3) − equals (x1 x2)(x2 x3)(x1 x3). − − − Formula. J 2 = a2b2 4a3c 4b3 27c2 + 18abc F . − − − ∈ An odd permutation of the roots takes J =(x1 x2)(x2 x3)(x1 x3) to J and an even permutation of the roots takes J to J. − − − − 1. Let f(x) F [x] be an irreducible cubic polynomial. ∈ (a). Show that, if J is an element of K, then the Galois group G(L/K) is the alternating group A3. Solution. If J F , then every element of G(K/F ) fixes J, and G(K/F ) must be A3, ∈ (b). Show that, if J is not an element of F , then the splitting field K of f(x) F [x] has ∈ Galois group G(K/F ) isomorphic to S3. -
The Kronecker-Weber Theorem
18.785 Number theory I Fall 2017 Lecture #20 11/15/2017 20 The Kronecker-Weber theorem In the previous lecture we established a relationship between finite groups of Dirichlet characters and subfields of cyclotomic fields. Specifically, we showed that there is a one-to- one-correspondence between finite groups H of primitive Dirichlet characters of conductor dividing m and subfields K of Q(ζm) under which H can be viewed as the character group of the finite abelian group Gal(K=Q) and the Dedekind zeta function of K factors as Y ζK (x) = L(s; χ): χ2H Now suppose we are given an arbitrary finite abelian extension K=Q. Does the character group of Gal(K=Q) correspond to a group of Dirichlet characters, and can we then factor the Dedekind zeta function ζK (s) as a product of Dirichlet L-functions? The answer is yes! This is a consequence of the Kronecker-Weber theorem, which states that every finite abelian extension of Q lies in a cyclotomic field. This theorem was first stated in 1853 by Kronecker [2], who provided a partial proof for extensions of odd degree. Weber [7] published a proof 1886 that was believed to address the remaining cases; in fact Weber's proof contains some gaps (as noted in [5]), but in any case an alternative proof was given a few years later by Hilbert [1]. The proof we present here is adapted from [6, Ch. 14] 20.1 Local and global Kronecker-Weber theorems We now state the (global) Kronecker-Weber theorem. -
January 10, 2010 CHAPTER SIX IRREDUCIBILITY and FACTORIZATION §1. BASIC DIVISIBILITY THEORY the Set of Polynomials Over a Field
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. A polynomials is irreducible iff it cannot be factored as a product of polynomials of strictly lower degree. Otherwise, the polynomial is reducible. Every linear polynomial is irreducible, and, when F = C, these are the only ones. When F = R, then the only other irreducibles are quadratics with negative discriminants. However, when F = Q, there are irreducible polynomials of arbitrary degree. As for the integers, we have a division algorithm, which in this case takes the form that, if f(x) and g(x) are two polynomials, then there is a quotient q(x) and a remainder r(x) whose degree is less than that of g(x) for which f(x) = q(x)g(x) + r(x) . The greatest common divisor of two polynomials f(x) and g(x) is a polynomial of maximum degree that divides both f(x) and g(x). It is determined up to multiplication by a constant, and every common divisor divides the greatest common divisor. These correspond to similar results for the integers and can be established in the same way. One can determine a greatest common divisor by the Euclidean algorithm, and by going through the equations in the algorithm backward arrive at the result that there are polynomials u(x) and v(x) for which gcd (f(x), g(x)) = u(x)f(x) + v(x)g(x) . -
Selecting Polynomials for the Function Field Sieve
Selecting polynomials for the Function Field Sieve Razvan Barbulescu Université de Lorraine, CNRS, INRIA, France [email protected] Abstract The Function Field Sieve algorithm is dedicated to computing discrete logarithms in a finite field Fqn , where q is a small prime power. The scope of this article is to select good polynomials for this algorithm by defining and measuring the size property and the so-called root and cancellation properties. In particular we present an algorithm for rapidly testing a large set of polynomials. Our study also explains the behaviour of inseparable polynomials, in particular we give an easy way to see that the algorithm encompass the Coppersmith algorithm as a particular case. 1 Introduction The Function Field Sieve (FFS) algorithm is dedicated to computing discrete logarithms in a finite field Fqn , where q is a small prime power. Introduced by Adleman in [Adl94] and inspired by the Number Field Sieve (NFS), the algorithm collects pairs of polynomials (a; b) 2 Fq[t] such that the norms of a − bx in two function fields are both smooth (the sieving stage), i.e having only irreducible divisors of small degree. It then solves a sparse linear system (the linear algebra stage), whose solutions, called virtual logarithms, allow to compute the discrete algorithm of any element during a final stage (individual logarithm stage). The choice of the defining polynomials f and g for the two function fields can be seen as a preliminary stage of the algorithm. It takes a small amount of time but it can greatly influence the sieving stage by slightly changing the probabilities of smoothness. -
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. -
Elements of Chapter 9: Nonlinear Systems Examples
Elements of Chapter 9: Nonlinear Systems To solve x0 = Ax, we use the ansatz that x(t) = eλtv. We found that λ is an eigenvalue of A, and v an associated eigenvector. We can also summarize the geometric behavior of the solutions by looking at a plot- However, there is an easier way to classify the stability of the origin (as an equilibrium), To find the eigenvalues, we compute the characteristic equation: p Tr(A) ± ∆ λ2 − Tr(A)λ + det(A) = 0 λ = 2 which depends on the discriminant ∆: • ∆ > 0: Real λ1; λ2. • ∆ < 0: Complex λ = a + ib • ∆ = 0: One eigenvalue. The type of solution depends on ∆, and in particular, where ∆ = 0: ∆ = 0 ) 0 = (Tr(A))2 − 4det(A) This is a parabola in the (Tr(A); det(A)) coordinate system, inside the parabola is where ∆ < 0 (complex roots), and outside the parabola is where ∆ > 0. We can then locate the position of our particular trace and determinant using the Poincar´eDiagram and it will tell us what the stability will be. Examples Given the system where x0 = Ax for each matrix A below, classify the origin using the Poincar´eDiagram: 1 −4 1. 4 −7 SOLUTION: Compute the trace, determinant and discriminant: Tr(A) = −6 Det(A) = −7 + 16 = 9 ∆ = 36 − 4 · 9 = 0 Therefore, we have a \degenerate sink" at the origin. 1 2 2. −5 −1 SOLUTION: Compute the trace, determinant and discriminant: Tr(A) = 0 Det(A) = −1 + 10 = 9 ∆ = 02 − 4 · 9 = −36 The origin is a center. 1 3. Given the system x0 = Ax where the matrix A depends on α, describe how the equilibrium solution changes depending on α (use the Poincar´e Diagram): 2 −5 (a) α −2 SOLUTION: The trace is 0, so that puts us on the \det(A)" axis. -
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) .