Normal Division Algebras Satisfying Mild Assumptions

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PROC. N. A. S.

to know only the index s and the first s + 2 terms of the sp sequence in order to write down the functional equations for the polynomials concerned. The foregoing adds a double infinity of such sequences to those already known. In conclusion we remark that the so-called "elementary method" of N. Nielsen and others is abstractly identical with the symbolic method of Blissard used in this paper. This fact has been misunderstood by some who, incidentally, erroneously attribute the method to Lucas. It is proved in my paper cited above that each method implies and is implied by the other.

NORMAL DIVISION ALGEBRAS SATISFYING MILD ASS UMPTIONS By A. ADRIAN ALBIRT DEPARTMENT OF MATHEMATICS, PRINCETON UNIVERSITY Communicated November 10, 1928

Let D be a normal division algebra in n2 units over an infinite field F. Every element of D satisfies some equation of degree n, with leading coefficient unity and further coefficients in F. There exists an infinity of elements of D satisfying an equation of this kind, irreducible in F. 1.

Definitions.

An element x of A shall be called of grade r if its minimum equation has degree r. An equation @+ ck(w)=r

a1w

+ ... + aop = 0

shall be called of type Rk if k - 1 distinct complex roots of +(w) = 0 are expressible as rational functions, with coefficients in F, of a kth distinct root. An element x of a normal division algebra D is said to be of type Rk if its minimum equation is of type Rk. A normal division algebra D in n2 units is said to be of type Rk if it contains an element x of grade n and type Rk2. Known Division Algebras. Let 2 be any associative division algebra over F. Let to every element A of 2 correspond a unique element A' of Z, and let s 5 0 be a self corresponding element of M. Let A' - (A')', AM' = (A")', .... LetA( = sAs1 for every A of Z. Let a' = a, (AB)' = A'B',(A + B)' =

VOL,. 14, 1928

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A' + B' for every a of F, and A and B of 2. Call any algebra 2 with these properties a base algebra. L. E. Dickson has shown that, given any base algebra 2, there exists an element z such that the algebra r, of elements

Ao + Alz + ... + Ap-, z§ and multiplication table

zA = A'z z'=s. where A ,A O ..., As,p range independently over all elements of 2, is an associative algebra over F. Hence, to every base algebra there corresponds a r algebra. Let SI be the set of all algebraic fields over F. All such algebras are known. Let T, be the set of all base algebras obtained by considering all possible self-correspondences of algebras of S. Let S2 be the set of all algebras which are division algebras obtained by considering the direct products of all r algebras obtained from T, and all algebras of Si, and taking all division algebras of this set. Considering all self-correspondences of the algebras of 52 we define the set T2. Continuing similarly we define the sets Sk for all k's. The totality of algebras Sk will be said to be known algebras and any algebra will be said to be determined if we can show that it belongs to one of the sets Sk We have proved the following theorems: THJsORZM 1. Let x be of grade r. Let x be of type Rk but not of type Rk +1* Then k is a divisor of r. TH1ORBM 2. Every normal division algebra in p2 units, p a prime, of type R2 is a cyclic (Dickson) algebra. THmORBM 3. Let D be the direct product of two normal division algebras B and C. Let B have order m2 and type Rm and C have order n2 and type Rn. Then, if D is a division algebra, it is a normal division algebra of type Rmn. THBORsM 4. Every normal division algebra in 4p2 units, p a prime, of type Rp is a known algebra. THI3ORBM 5. Every normal division algebra in 36 units of type R2 is of type R13. This theorem shows that all normal division algebras in 36 units of type R2 are known algebras. Since every normal division algebra is obviously of type R,, this is a very weak assumption. The algebras determined by L. E. Dickson,' in 4p2 units, as a generalization of the author's algebras in 16 units, were algebras essentially containing an element of grade 2p satisfying an equation with even powers, only and such that if c(w2) = 0 is the equation then +(p) = 0 has the

PROC. N. A. S.

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cyclic group with respect to F. The author has shown that the algebras in 16 units were Cecioni algebras of type 1?4 and, in general, THZORM 6. Let D be a normal division algebra, over F in 4p2 units, p a prime. Let D contain x, of grade 2p satisfying

(W) _(A)2p + alo2(P- 1) + ... + ap = 0 such that s&(c) =,(c2) and +(p) is cyclic with respect to F. Then A is a known algebra of type R2p. All normal division algebras in n2 units of type R&, such that x of type R, satisfies a normal equation with solvable group, have been shown to be known algebras and have been constructed.2 No known normal division algebras not equivalent to algebras of the aforesaid kind have been shown to exist, but this existence question is a construction problem, not a determination problem. 1 Bulletin of the American Mathematical Society, vol. XXXIV, 5, Sept.-Oct., 1928, p. 555. This paper was at the printers before it was shown that the 16 unit algebras were of type R4. 2 L. E. Dickson, "New Division Algebras," Transactions of the Am. Math. Society, 28, pp. 207-234, April, 1928.

THE GROUP OF THE RANK EQUATION OF ANY NORMAL DIVISION ALGEBRA By A. ADRIAN ALBERT1 DEPARTMIONT OF MATHZMATICS, PRINCZTON UNIVURSITY

Communicated November 10, 1928

Let K be any infinite field. The Hilbert "Irreducibility Theorem" states the following: Consider the equation:

f(x, 1, .. .,Ix)

x' + Fi(Xi, ... i Xr)x'

+

+

...(Xr) F(X19

0

(1) where Fi(XI, ..., k) are rational functions, with coefficients in K, of the independent parameters Xi, . . ., 'X. Let the group of f(x) with respect to K(X1, ..., )xr) be r. Then there exist an infinitude of rational values of the parameters X1, . . ., X,r, such that the resulting numerical equation has the group r with respect to K. Using a proof following the lines of Hilbert's proof of the above theorem we have shown the truth of the following general theorem. THJWRoM 1. Let f(x; X1, ... ., h) = 0 be an equation (1) with group