A Geometric Approach to Classical Galois Theory R.C. Churchill Hunter College and the Graduate Center of CUNY, and the University of Calgary August 2010 Prepared for the Kolchin Seminar on Differential Algebra Graduate Center, City University of New York 27 August 2010 Contents §1. Introduction §2. Preliminaries on Cyclic Vectors §3. Preliminaries on Diagonalizability §4. Galois Extensions §5. The Galois Group §6. Fundamental Matrices §7. Preliminaries on Permutation Matrices §8. Preliminaries on Semi-Direct Products §9. Faithful Matrix Representations of the Galois Group Notes and Comments References Throughout the notes K denotes a field, n is a positive integer, and V denotes an n-dimensional vector space over K. The characteristic and minimal polynomi- als of a K-linear operator T : V → V are denoted charT,K (x) and minT,K (x) respectively: both are elements of K[x]. The order of a finite group G is denoted |G|. 1 1. Introduction There are striking parallels between standard techniques for studying the zeros of single-variable polynomials with field coefficients and standard techniques for studying P dj the zeros of ordinary linear differential operators j fj(x) dxj on appropriate fields of functions. These become even more apparent when the latter are formulated algebraically, and that is our first item of business. Let R be a ring, not necessarily commutative, with unity (i.e., multiplicative identity). A function δ : r ∈ R 7→ r 0 ∈ R is a derivation (on or of R ) if the following two properties hold for all r1, r2 ∈ R: 0 0 0 • (“additivity”) (r1 + r2) = r1 + r2, and 0 0 0 • (“the Leibniz rule”) (r1r2) = r1r2 + r1r2. Note from the Leibniz rule and induction that Qn 0 P 0 (1.1) ( j=1 rj) = j r1r2 ··· rj−1rj rj+1 ··· rn for any n ≥ 2 and any r1, r2, . , rn ∈ R. A pair (R, δ) as in the previous paragraph is called a differential ring, or a dif- ferential field when R is a field, although when δ is clear from context one generally refers to R as the differential ring (or field, as the case may be). Indeed, the custom is to minimize specific reference to δ, within reason, by writing δnr as r(n) for any non-negative integer n and any r ∈ R, where δ0r := r. For small n > 0 primes are also used, e.g., r 00 := δ2r = (r 0) 0 and r 000 := δ3r = (r 00) 0. By regarding an element r ∈ R as the left multiplication function µr : t ∈ R 7→ rt ∈ R one can view R as sitting within the ring FR of (set-theoretic) functions f : R → R, and FR is thereby endowed with the structure of an R-algebra. The derivation δ is obviously an element of FR, and as a result one can consider the R-subalgebra R[δ] of FR generated by R and δ. Any element of R[δ] can be expressed in the “polynomial” 1 Pm j form j=0 rjδ , and these are our generalizations of “linear ordinary differential operators.” The “parallels” alluded to in the opening paragraph should henceforth be regarded Pn j as being between (ordinary) polynomials p = j=0 rjx ∈ R[x] (with no derivation Pn j assumed on R), and ordinary linear differential operators L = Lp = j=0 rjδ ∈ R[δ]. 1This form will be unique under quite mild assumptions which we will simply assume are satisfied, and, to avoid a lengthy digression, will not state explicitly. Uniqueness holds, for example, when R d is the field C(z) of rational functions (quotients of polynomial functions) and δ = dz . 2 However, before beginning that discussion it might be a good idea to point out one major difference between the R-algebras R[x] and R[δ], i.e., that the former is commutative (as readers are surely aware), while the latter is generally not (as readers may not be aware). Indeed, for any r, t ∈ R one sees from the Leibniz rule that (δ ◦ µr)(t) = δ(µr(t)) = δ(rt) = r δt + δr t = (µr ◦ δ)(t) + δrt = (µr ◦ δ + µδr)(t), and we therefore have (1.2) δ ◦ µr = µr ◦ δ + µδr. In keeping with blurring all distinction between r and µr, and minimizing specific references to δ, this would generally be written (1.3) δr = rδ + r 0 for all r ∈ R. A derivation r ∈ R 7→ r 0 ∈ R is trivial when r 0 = 0 for all r ∈ R, and in that case we see from (1.3) that R[δ] is commutative. Indeed, under that hypothesis the correspondence x ↔ δ induces an isomorphism of the R-algebras R[x] and R[δ] which one learns to exploit in a first course on ordinary differential equations, wherein d one generally assumes that R = R or C and that δ = dx . In that context the first Pn j hint of any parallelism between polynomials p = j=0 rjx and the corresponding Pn j linear operators Lp = j=0 rjδ takes the form of the observation that for any λ ∈ R λx λx λx one has Lpe = p(λ)e , hence that for any root λ of p the function y = e is a th solution of the n -order equation Lpy = 0. For the remainder of the introduction R denotes a (commutative) integral domain with unity 1 = 1R. For our purposes the deeper analogies between polynomials and linear differential operators are most easily revealed by treating the former in terms of Vandermonde matrices and Vandermonde determinants. For ease of reference, and to establish our notation, we recall the definitions. 3 Suppose n ≥ 1 and r1, r2, . , rn are (not necessarily distinct) elements of R[x]. n The classical Vandermonde matrix of r := (r1, r2, . , rn) ∈ R[x] is defined to be the n × n matrix i−1 (1.4) vdmmR[x],n(r) := (rj ) ∈ R[x], which is sometimes more conveniently expressed as vdmmR[x],n(r1, r2, . , rn). Using full matrix notation the definition is: 1 1 ··· 1 r r ··· r 1 2 n . (1.5) vdmmR[x],n(r) = vdmmR[x],n(r1, r2, . , rn) := . , rn−2 rn−2 ··· rn−2 1 2 n n−1 n−1 n−1 r1 r2 ··· rn The associated determinant (1.6) vdmdR[x],n(r) = vdmdR[x],n(r1, r2, . , rn) := det(vdmmR[x],n(r)) is called the Vandermonde determinant of r = (r1, r2, . , rn). n+1 When r ∈ R[x] has the form (t1, t2, . , tn, x), with tj ∈ R for j = 1, . , n, it proves convenient to first express the (n + 1) × (n + 1) Vandermonde matrix vdmmR[x],n+1(r) as vdmmR,n+1(t; x) = vdmmR,n+1(t1, t2, . , tn, x), and to then re- label t as r and all tj as rj. In other words, vdmmR,n+1(r; x) = vdmmR,n+1(r1, r2, . , rn; x) := vdmm (r , r , . , r , x) R[x],n+1 1 2 n 1 ··· 1 1 r ··· r x (1.7) 1 n . = . . n−1 n−1 n−1 r1 ··· rn x n n n r1 ··· rn x The associated Vandermonde determinant is expressed accordingly, i.e., (1.8) vdmdR,n+1(r; x) := vdmdR[x],n+1(r1, r2, . , rn, x) := det(vdmmR,n+1(r; x)). 4 Proposition 1.9 : Assume n ≥ 1, choose any (not necessarily distinct) elements n r1, r2, . , rn ∈ R, and set r := (r1, r2, . , rn) ∈ R . Then the following assertions hold. Q (a) The Vandermonde determinant vdmdR[x],n(r) has value i>j(ri − rj). Q (b) The determinant vdmdR,n+1(r; x) has value vdmdR[x],n(r) · j(x − rj). (c) vdmdR[x],n(r) 6= 0 if and only if the elements rj are pairwise distinct. Pn j (d) Suppose R is a field, n ≥ 1, p = j=0 ajx ∈ R[x], and an 6= 0. Then p has at most n roots in R. If the number of distinct roots of p in R is k ≥ 1, then a collection of roots t1, t2, . , tk ∈ R of p is precisely that collection of distinct roots if and only if vdmdR[x],k(t1, t2, . , tk) 6= 0. (e) The determinant vdmdR,n+1(r; x) is a degree n polynomial in R[x] having r1, r2, . , rn as a complete set of roots. Moreover, when vdmdR[x],n(r) is a unit of R the rj are pairwise distinct and the product −1 (i) vdmdR[x],n(r) · vdmdR,n+1(r; x) is a monic degree n polynomial in R[x] having precisely these roots. (f) Suppose p ∈ R[x] is a monic polynomial of degree n ≥ 1 having r1, r2, . , rn as a complete set of roots and vdmdR[x],n(r) is a unit of R. Then p must be the polynomial defined in (i) of (e). n Pn−1 k (g) Suppose r1, r2, . , rn are the zeros of a monic polynomial p = x + k=0 akx ∈ R[x]. Then (ii) θ(vdmdR[x],n(r)) = −an−1 · vdmdR[x],n(r), where θ(vdmdR[x],n(r)) denotes the determinant of the “modified” Vander- monde matrix 1 1 ··· 1 r r ··· r 1 2 n . . , rn−2 rn−2 ··· rn−2 1 2 n n n n r1 r2 ··· rn 5 (h) (Reduction of Degree) Suppose r, s ∈ R, r is a zero of a polynomial p = Pn k k=0 akx ∈ R[x], and s 6= 0. Then r + s is a zero of p if and only if s is a zero of the polynomial ! ! k (iii) Pn−1 Pn a rk−1−j xj.