26 A2ATHEMATICS: H. S. VANDIVER PROC. N. A. S. of A. Cayley. On the other -hand, W. R. Hamilton did not use the term group but favored the use of his own term quaternions for the development of a theory based upon the special abstract group of order S which-involves 6 operators of order 4. The associative law was later very commonly embodied in the definitions of an abstract group and the development of these groups did much to emphasize the fundamental importance of this law. These observations in regard to W. R. Hamilton seem to imply that the theory of abstract groups may reasonably be regarded as about ten years older than the assumption that A. Cayley was the founder of this theory would imply. Implicitly, W. R. Hamilton emphasized the importance of the quaternion group in his Lectures on Quaternions, page xxvi (1853). LIMITS FOR THE NUMBER OF SOLUTIONS OF CERTAIN GENERAL TYPES OF EQUATIONS IN A FINITE FIELD By H. S. VANDIVER DEPARTMENT OF APPLIED MATHEMATICS, UNIVERSITY OF TEXAS Communicated May 22, 1947 Dickson,I using cyclotomic integers, obtained an inferior limit for the number of solutions of the congruence xe + ye + ze 0 (mod p) (1) where e and p are given primes with xyz 0 (mod p). Hurwitz2 gave = superior and inferior limits for the number of solutions of the congruence axe + bye + czc O (mod p), (2) where abcxyz X 0 (mod p). Recently, the writer has considered generalizations of these results and has arrived at limits both superior and inferior, for the number of solutions of the equation ClXla + CXa2 +... + C,Xs + cs+ = 0 (3) in the x's, where the a's are integers such that 0 < a. pt-1; s > 2; the c's belong to a finite field of order pt, p prime, which will be designated by and F(p'); C1C2... CsXlx2... Xs $ 0 in F(p'). As a corollary to this it is possible to show that if we take the c's in (3) as rational integers and put t = 1, so that we have a congruence modulo p in effect, then for p sufficiently large the congruence has solutions,
VOL. 33, 1947 MATHEMATICS: H. S. VANDIVER 237 with the x's all prime to p. There is a similar theorem involving algebraic numbers. However, in the present paper we shall confine ourselves to the derivation of certain quadratic relations between the numbers of solutions, xm, yi, of trinomial equations of the type 1 + axm = bytm (3a) for various values of a and b in F(pt), abxy 0, 'as given in Theorem I. Other quadratic relations connecting such solutions have been given before,3' 4 but those given here are of a quite different character, and they will be used in another paper in the proofs of the results concerning (3) referred to above. Consider a finite field of order pt designated by F(pt), where p is a prime. Write pt = 1 + mc. Let g be a generator of the cyclic group formed by the non-zero elements of F(pt). Further let (i, j) denote the number of solutions gt, gs of 1 + gi+rm = gj+s(3b) if r and s are each in the range 0, 1,..., c - 1, noting that gmc= 1. If we write ind. for index and gtndx = x, represent the index of (-1), modulo m, by e, then for any i and j it is known4 that (i,j) = (j+e,i+e) =(-j+e,i-j) = (i-j+e, -j) = (-i, j - i) = - i + -i+e). (4) Also we have4 (i, 0) = c-1; (i, j) = c, (4a) s i where i 0, 1,...,m- 1; j = 1, 2,...,m- 1 modulo m. We also note that (i, j) = (i + am, j + Om) for any integers a and 13. A fundamental relation we shall employ in what follows is (Mitchell,4 his 4I' function defined on p. 165; for b = a, a + b d) where 4a, d(af) a, d(al) = Pt (4b) "a, d( (a = (i, j)-ai+dj i,j a being a primitive mth root of unity, a and d any integers subject to the conditions a W 0, d W 0, a - d 0 0(mod m) with i and j ranging independently over the integers 0, 1,..., m - 1.
238 MATHEMATICS: H. S. VANDIVER PROC. N. A. 8. We shall now determine the quantities O to m-1 z (i, j) (i +. h, j + k) = Ahk (5) i, j First we determine Aoo. Now if a g 0 (mod m) d 0 0 (mod m), d - a 0 0 (mod m) then if a is a primitive mth root of unity, (4b).gives 0 to m_l Ei hijk (i,j) (i+h,j+k)ah+dk = t (6) Let pt = 1 + mc. Consider the summation m-1 Otom-1 m-1 E (i,j)(i + hi,j + k)c _ah+dk = y Cad (7) d=o hijk d=o For a 0 O(mod m), d 0 O(mod m), a - d g O(mod m) each term in the sum with respect to d = pt. Now consider the value when d = 0, which is 0 to m-1 E hijk (i,j)(i + h,j + k)a-ah Set i + h = i', j + k = j' then this expression becomes i,j,,$ This is obviously equal to Z (i, j)(i', jd)a W - i)a = Cao (8) if O to m-1 and each of these equals (-1) if a ) Oto m-1i ( 7 (ij)aia (i,j, /a-i'a (8a) 0(mod m), hence Cao=1 (9) Now consider the case when a - d =O(mod m) in (7). term reduces to or (i, j) (i,ji)aa(h-k) = Ca, E(i, j) a- ali -j)e(i,, y) C - a(i" If i - j = f, then for a 0 O(mod m), using (4), j,.f,(f + E -j)caaf = (f -j ) af - and similarly for the second factor. j,f C. = 1, a O(mod m). t The corresponding (10)
VOL. 33, 1947 MATHEMATICS: H. S. VANDIVER.239 Now if a 0=(mod m) in (8) then we find - Coo= (mc -1)2 (11) employing (8a) and (4a). We now simplify (7) under the assumption that a (7), (8), (9) and (10), we find 0 (mod m). Using d=0 Cad (m -2)p + 2; a O(mod m). (12) For the case where a = 0(mod m) we obtain in -the same way that (9) was derived Cw =1,d 0(modm). (13) rn-i Cod (mc-1)2 +m 1 (13a) d =0 Now the left hand member of (12) may be written d-o where Ahk is defined as in (5). Now rn-1 or from (12), hk AMhka) Ahka + d-o hk a.+- - Z Ahka ah(1±+ ak + + (rn1)k hk Oif k 0 (mod m). d-o ~ Ahak "ah+dk = iaia-ah hk me Ah,oaGk = (m -2)pt + 2, (14) h if a 0 (mod m) and it equals (mc-1)2 + n-l otherwise. mn-i mahoch = (m -2)(mc + 1)(m - 1) + 2(m - 1) + a=o (mc-1)2 + m-1
or A20ATHEMATICS: H. S. VANDIVER 240 PROC. N. A. S. or m2aoo = m2c2 + m3c - 3m2c + m2 Aoo =C(c12 + (m-l)c Now for a 0 (mod m) we have from (14) meaoa +a = asa((m - 2)pt + 2) h and for a 0 (mod m), mzah,= (mc-i)2+m-l. h (14a) rn-i m E EAha -ah+sa = (mc- 1)2 + m- 1- ((m -2)pt + 2) a=o h (C2 -C)M, or m2ato = m2(c2 -C) A,w = c2-c, h O (mod m). (15) We shall now show also that We have But by (1) Aok = c- c, k 0 0 (mod m). (16) Aok = (i, j)(i,j + k) ".7 (i, j) = (j+,i+ e) (i,j+k) = (j+ k + e,i+,e) j (i, j) (i, j + k) = Z i, j7j+e, i+e ((j + e) + k, i + e) (j + e, i + e), where j + e and i + e range over the same set 0, 1,... - 1, modulo m as do i and j so that this equation gives (16), using (15). We lastly determine Ahk with h 0, k 0 0 (mod m). Consider the sum m-1 Oto m-1 rn Z (ij)(i + hj, + k)cjah+dkvd = (17) d =O hijk We know from (6) that each term corresponding to a given d equals p'a-vd except when d 0, a 0, or d _ a, (mod m). For d- 0, then
VOL. 33, 1947 MATHEMATICS: H. S. VANDIVER 241 the corresponding term equals unity if a 0 (mod ni), using (9). For d = a we have by (10) that the term is a-a. B = 1 + -Vpt + a-2v pi +... + a-(mr-l)v pt + -av _ -av pt Bd= 1 pt + a,(1 _ pt) (18) But Bad, can also be written as rn-i MAhva ah + E Ahka ah(l + ak-v + + a((m-1)(k-v)) k =O k#v and the last summation is 0. by (18) mahvaa-ah = 1 - pt + -av(l - pt) or and Ah,aah = -_-C a avc (19) Ah,a = cas- Ca-av+sa (20) We also have, from (17), for a = 0 (mod m) Bod, = Mc2-2c (21) for if we take (17) with a =- 0 (mod m) then for d 0 (mod m), we find that the corresponding term is (mc - 1)2 while the other terms, corresponding to each d are a-v, a- 2v...a (m-1)v, using (13). From (20) and (21) we find m-1m- a=o = - -h+s cs2 Aaah+sa (_caa Ca +s) + M2 2C MC2, a=o if v W s (mod m), or As,= c2; s W O, v 00, s - For v v (mod m). (21a) = s we have from the above A8S = c2- c; s 0 0 (mod m). (22). * we have, employing (14a), (15), (16), (21a) and (22), THEOREM I. Set 0 to m-1 Ahk = (i,j)(i + h,j + k), i,j where (i, j) is the number of -sets of values f and I in the set 0, 1,..., c - 1, which satisfy the equation
242 MATHEMATICS: T. Y. THOMAS PROC. N. A. S. 1 + gi+fm=g + m in the finite field F(pt) where p is a prime such that pl = 1 + cm, g being a primitive root in F(pt). Then Aoo = (c-1)2 + c(m-1); (23) Aho = Aok = c2 c, h O, k O (mod m); (24) Ahk = c2, with h 0 k (mod m); h 0 0, k 0 0 (mod m); (25) Ahk = C2 - C, (26) if h-k (mod m). 1 Crelle, 135, 181-188 (1909). - Cf. also Pellet, Bull. Math. Soc. France, 15, 80-93 (1886). 2 Crelle, 136, 272-292 (1909). 3 THESE PROCEEDINGS, 32, 47-52 (1946). 4 Mitchell, Proc. Amer. Math. Soc., 17, 167 (1916). CHARACTERISTIC COORDINA TES FOR HYPERBOLIC DIFFERENTIAL EQUATIONS IN THE LARGE By T. Y. THOMAS DEPARTMENT OF MATHEMATICS, INDIANA UNIVERSITY Communicated May 31, 1947 Partial differential equations of hyperbolic type are habitually treated by the introduction of characteristic coordinates. When such equations occur in physical problems their solutions in the large are necessarily demanded. In spite of this fact the existence of characteristic coordinates appears to be established only locally. We shall here give conditions for the existence of a (1, 1) differentiable transformation defining the characteristic coordinate system in the large. The form of these conditions is suitable for practical application but such applications of the theorem will not be included in this communication. Let D be an open simply connected two dimensional domain, finite or infinite, referred to a system of rectangular coordinates xa. At each point of D characteristic directions X are defined as the solutions of the equation = 0 in which the summation is over the values 1, 2 of the indices and it is supposed that the coefficients gag3 are continuous and have continuous first partial derivatives in D. Assuming det. lg,df < 0 (hyperbolic case) there will be exactly two distinct characteristic directions at each point of D and these directions will generate two families or congruences of characteristic curves each of which will, cover D completely. In fact-any point P of D is contained in a local co6rdinate system, e.g., a