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The Galois Group of a Polynomial Equation with Coefficients in a Finite Field

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The Galois Group of a Polynomial Equation with Coefficients in a Finite Field University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange Masters Theses Graduate School 8-1950 The Galois Group of a Polynomial Equation with Coefficients in a Finite Field Charles Leston Bradshaw University of Tennessee - Knoxville Follow this and additional works at: https://trace.tennessee.edu/utk_gradthes Part of the Mathematics Commons Recommended Citation Bradshaw, Charles Leston, "The Galois Group of a Polynomial Equation with Coefficients in a Finite Field. " Master's Thesis, University of Tennessee, 1950. https://trace.tennessee.edu/utk_gradthes/3246 This Thesis is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Masters Theses by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a thesis written by Charles Leston Bradshaw entitled "The Galois Group of a Polynomial Equation with Coefficients in a Finite Field." I have examined the final electronic copy of this thesis for form and content and recommend that it be accepted in partial fulfillment of the equirr ements for the degree of Master of Arts, with a major in Mathematics. Robert L. Wilson, Major Professor We have read this thesis and recommend its acceptance: G. E. Albert, O. G. Harrold, Jr. Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School (Original signatures are on file with official studentecor r ds.) July 31, 1950 To the Gradua te Council : I am submi tting he rewith a thesis written by Charles Leston Br adshaw enti tl ed "The Galois Group of a Polynomial Equation with Coefficients in a Finite Fi eld." I recommend tha t it be accepte d for six quarter ho ur s of credit in par­ tial fulfillment of the requirements for the degree of Ma ster of Ar ts , with a ma jor in Mathemati cs. ��--- ia'jor Pr ofessor - We have read this thesi s and recomme nd its acceptance : � () flr�J----;.. - -· Accepted for the Council : �ttL� - �� the Graduate chool THE GALOIS GROUP OF A POLYNOMIAL EQUATION WITH COEFFI CIENTS IN A FINITE FIELD A THESIS Submitted to The Commi ttee on Graduate Study of The Un iversity of Tennessee in Partial Fulfillment of the Requirements for the degree of Master of Arts ----- by Charles Leston Bradshaw August 195'0 ACKNOWLEDGMENT The author wi shes to expre ss his appre ciation to Professor R. L. Wi lson , under whose direction this thesi s was wr itten , for hi s suggesti on of the topi c and for his valuable assistance in its preparation. TABLE OF CONTENTS CHAPTER PAGE I. INTRODUCTION • • . • • • • • . • • 0 0 0 • 0 1 II. DEFINITIONS AND NOTATIONS • • • • • • • • • • 0 4 III . DETERMINATION OF THE GALOIS GROUP •• 0 • • • 0 9 IV. EXAMPLES • . 0 0 Q 0 0 Q 0 23 -·- •• . • 0 0 • • v. SUMMARY . 39 CHAPTER I INTROD UCTION The primary purpose of this thesi s is to demonstrate a method for the determination of the Ga lois group of a poly­ nomial equation with coefficients in a finite field. The problem of finding the Ga lois group of an arbi trary equation, where the coefficient field is either finite or infinite, is 1 neither new nor unsolved . Van der Waerden illustrates a method which is dependent upon the fact that the permutations of the Ga lois group are characterized by the property that they transform the quantity Z = x1r 1 + x2r2 + · · �· + xnrn into its conjugates. The xi denote n inde terminants �nd the ri the roots of the polynomial under consideration. He performs all poss ible permutations of the indeterminants � on Z and forms the pr oduc t F(y ,x) = r:r( y - sxZ), where the � de note the permutati ons or the x1. It is then proved that if the polynomial F(y, �) be decomposed into irreducible factors, the permutations sx whi ch earry any on e of the irreduci ble !actors into itself is exactly the Galoi s 2 group of the given equation. Ca jori suggests a method, l a. L. van der Wa erdent Modern Al gebra (Bew York: Frederick Un gar Company, 1949J, p. 189. 2 F. Ca jori, An Introduction to �he MQdern Theory ot Egua tion! (New York : The Macmillan Company, 1904}, p. 169. 2 somewhat similar to this, in which he forms the Galois re­ solvent and tests it for reducibility. Wilson3 completely solves the problem in the case of the infinite field. The method used in this thesi s follows that of his in so far as it is applicable. The me thod as given here is dependent upon the determi­ nation of the roots of an induc ed equation. This method is used to determine suc cessively whether the Galois group is or is not contained in each of the subgroups of the symmetric group of degree n, where n is the degree of the equation under consideration . The information so obtained permits us to determine which of these subgroup s is the Galois group, or whether the Galois group is the symmetric group. We need only note which subgroup contains the Galois group but has no sub­ group containing the Galois group . We shall first se lect an arbitrary subgroup of the symmetric group of degree n, where again n is the degree of the given equation. Having selected one suc h group it will be necessary to display a rational function of the roots of the given equation with rational coefficients, which is in­ variant und er all of the permutations of the selected group. We then construct an equation which has coefficients in the coefficient field of the original equation, and which has this 3R. L. Wilson, "A Me thod for the Determination of the Galois Group.n To appear soon in the Duke ¥ath§mA�a l Journal. 3 function as one of its roots. We call the equation so con­ structed the induced equation. It is now necessary to deter­ mine if the function displayed above , or one of its conjugates, is an element of the coefficient field �. If we find that the induced equation has no roots in F, then surely the function is not in F. It may happen that all of the roots of the induced equation, which are in F, are multiple roots. In this case we show , by theorem 8, that it is possi ble , in a finite number of steps, to construct an induced equation which has no roots in the coefficient field , or else to conclude that the Galois group is conta ined in the se lected subgroup. In most instances it will be unnecessary to use theorem 8. We are now in a position to apply a basi c theorem in Galois theory (theorem 5). In case the function of the roots, which we have constructed , is a non-repeated root Qf the in­ duced equa tion and is in F, the Galois group is contained in the subgroup we had initially se lected . If the function is not in F, then the group we have se lected does not contain the Galois group. Examples are given in Chapter IV illustrating this method for finding the Galois group. CH APTER II DEFINITIONS AND NOTATIONS We state the following well-known definitions and theorems . DEFINITION 1. A field E is a system of elements closed under two binary operations, addition and multiplication, such that (a ) und er addition, E is a commutative group with id entity zero; (b) under multiplication, the elements of E not zero form another commutative group; (c ) addition is distributive with respect to multiplication. DEFINITION 2 . If there is a positive integer � such that p •a = 0 for each � in a field F, and if � is the smallest integer with this property , then F is said to have cha!acteri�tic � · If no such element exists then F is said to have char�ristic � · The two types of fields are often referred to as finite and in finite, respectively. DEFINITION 3. A set of linearly independent quantities ul, u2 , · ··,un of an extension � over E is said to form a basis of � over E if each �, which is an element of R can be expressed as a linear combination ; x = a1u1 f a2u2 f ••• + anlln' where a 1, a2, ···, an are elements of F. DEFINIT ION 4. An extension N of F is called a !QQ1 fiel� of a polynomial f(x) of degree n wi th coefficients in F if (a ) f(x) can be factored into linear factors in N, and; (b) N can be generated over F by the roots of f(x), as N = F(r 1, r2, ••• , rn)• DE FINITION 5. If N = F(r 1, r2 , •••, rn) is the root field of a polynomial f(x) = (x - r 1)(x - r2) ••• (x - rn), . then the automorphism group of N over F is-known as the Galois group of the equa tion f(x) = 0 or as the Ga lois group of N over F. DEFINIT ION 6. A polynomial f(x) of degree n is called sep�able over a field F if there exists n distinct roots of f(x) in some root field N ? F; otherwise f(x) is called inseparabl§. 1 THEOREM 1. The number � of elements in a finite field of characteristic Q (p must be prime) is a power of the prime Q, that is, s = pn. lR. D. Carmichael, In troduc tion to the Theory of Groups of Fiq!te Qrde� (Boston: Ginn and Company, 1937� p.
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