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This paper gives the first primitive polynomial of the third kind of degree n over GF ( p d ) {\text {GF}}({p^d}) for each p, d, n satisfying p > 10 2 , p d > 10 3 , p d n > 10 6 p > {10^2},{p^d} > {10^3},{p^{dn}} > {10^6} .

For T2 spaces, it is shown that any left monotone partial order on a connected space with a finite primitive dispersion set has a minimal element. Also, any non-void closed left monotone transitive relation on a connected space with compact primitive dispersion set has a minimal element.

The following list contains one example of a primitive polynomial (mod 2) for each degree », 1 ^ n ^ 100. It was compiled with the aid of the Mercury computer at Manchester University by the following method. The polynomials P„(x) (mod 2) of degree n were tested in their natural order until a primitive polynomial was found. The test comprised three stages. In the first stage the small primes, of degree up to 9, were tried as possible factors (mod 2) of Pn . If no factor was found P„ went forward to the second stage, which tested whether Pn divides x" — 1, where N = 2" — 1. If it does, and N is prime (a Mersenne prime), this suffices to prove that P„ is primitive. If N is composite, however, Pn might divide xM — 1, where M is a factor of N, and then Pn would not be primitive. The third stage was, therefore, a trial of this possibility, in which M took the values N/p, where p runs through the prime factors of N. The two latter stages were carried out by a process in which the computer repeated the operations of squaring, possibly multiplying by x (depending on the binary representation of M), then dividing by Pn . The prime factors of N were taken from the tables of Kraitchik [1], supplemented by Robinson's [2] further decomposition of 296 — 1. If any more of these 'prime' factors should turn out to be composite, doubt would be cast on the corresponding Pn . Mersenne polynomials for n = 107 and 127 are also given. The prime x127 + x + 1 was found by Zier 1er [3]. Its nature follows from the general result that if ?,anxn divides 2c„x" (mod p), then

By introducing the concept of a knice submodule, a refinement of the notion of nice subgroup, we are able to formulate a version of the third axiom of countability appropriate to the study of p p -local mixed groups in the spirit of the well-known characterization of totally projective p p -groups. Our Axiom 3 3 modules, in fact, form a class of Z p {{\mathbf {Z}}_{\mathbf {p}}} -modules, encompassing the totally projectives in the torsion case, for which we prove a uniqueness theorem and establish closure under direct summands. Indeed Axiom 3 3 modules turn out to be precisely the previously classified Warfield modules. But with the added power of the third axiom of countability characterization, we derive numerous new results, including the resolution of a long-standing problem of Warfield and theorems in the vein of familiar criteria due to Kulikov and Pontryagin.

For a field K, let K denote its algebraic closure. Assume that IK: KI = oo. Then for any linear groups G, H C GL (K) their free product G * H can be embedded into GLN(K(t)). Here N is an integer depending on K only and t stands for an indeterminate. If K is a field, let K denote its algebraic closure. We are going to prove THEOREM. Let G, H C GLn (K) be subgroups such that all entries of matrices g E G, h E H lie in a subfield Ko C K with IKo: Kol = ox. Then the free product G * H can be embedded into GLN(K(t)). Here t is an indeterminate and N is an integer depending on Ko and n only. We need two lemmata. LEMMA A. Let L be a field satisfying IL :LI = ox. For any integer m there exists a E L such that degL a > m. PROOF. If L is separable over L it follows from the Theorem of Primitive Element [1, Chapter VII, ?6, Theorem 14, p. 185]. If not, i.e. if L is not perfect, say a E L\LP, then the elements al/p, al/P,2. ... have degrees p, p2, ... over L. D LEMMA B. LetL be afield satisfying IL: LI = ox. If S C GLn(L) is a subgroup with S n L * ln = {ln,} then there exists an embedding S C GLn(L) such that (i) IL: LI n over the previous field L C L(all) C L(all, a12) C . C L(all,... ,a) = L Before we conclude the proof let us make a REMARK. If p E L[xij] is a nonzero polynomial and degp 91,2n+1 #: 0, (iii) h EH\111 =:>h2n+l,l 780. These are precisely the assumptions of Shalen's Theorem [2, Proposition 1.3]. It follows that the map p: G * H -* GL2n+1(KO[t, t']) defined by 0] [1 01t-2 G9gt, . .9. . ~~~, H9h-h, ? ~~t2n+ 1 Ot-(2n+l has the following property: If a = p(gi)p(hi) * * . p(gr)p(hr) and all gi, hi :A 1, then Trace(a) = a. t2nr+ lower (possibly negative) powers of t; a + 0. Clearly, p is an embedding of G U H. Further, each element w of G * H\(G U H) is conjugate to one in the form g1hl * *. grhr with gi E G\{1}, hi E H\{1}. Therefore, Tracep(w) $8 Tracep(1), hence p(w) $& 1. Thus p embeds G * H into GL2n+1(Ko(t)). To finish the proof, notice that Ko(t) = Ko(t)(aij) and so Therefore, IKo(t): Ko(t) = d < oo. G * H C GL(2n+l)d(K0(t)) C GL(2n+1)d(K(t)). D This content downloaded from 157.55.39.238 on Tue, 22 Nov 2016 04:07:23 UTC All use subject to http://about.jstor.org/terms

Primitive polynomials over GF(2) are used for generating maximal-length sequences. Powers of a root α of such a polynomial are delay operators on the generated sequence. Given two primitive polynomials of degree n with α1, and α2 their roots, it is shown which parameter is needed in order to find α1x from α2x without knowledge of x.

A Quick method is developed to find an element or order 2p+1p in the finite field GF(q2), where q = 2P−1 is a Mersenne prime. Such an element is needed to implement complex integer transforms of length 2kp over GF(q2 where 3 ≤ k ≤ p + 1.

3 -4u2 + 4u + 1 _ (2u + 1)2 (mod p). This gives us an explicit form of the square roots of -3 modulo p. (Note that -3= (2w+ 1)2 where w is a complex cube root of 1 other than 1.) Now let p 6n+5 be a prime and let b be a primitive element modulo p. Suppose that -3 is a quadratic residue modulo p, then there is an element c such that c2-3 (mod p). Define u by the congruence c 2u + 1 (mod p), then we get that u2+u+1=0 (mod p) whence u3 =1 (mod p). Let k be an integer such that bk-u (mod p), then b3ku U3-1 (mod p) so (p-1) I (3k). Since p-I (=6n+4) is relatively prime to 3, it follows that (p-l)jk whence 1-bk-u (mod p). Thus c=2u+1=3 (mod p), consequently 12-c2+3-0 (mod p), which is a contradiction. Therefore -3 is a quadratic nonresidue modulo p. (A contradiction can also be obtained by observing that u 1 (mod p) and u2+u+ 1 (mod p) imply 3 0 (mod p). This was suggested by Professor George Whaples.) The same method also serves to establish the quadratic character of -1

We show that if a Hopf algebra has finite dimensional primitives and a primitive lies in arbitrarily long finite sequences of divided powers then it lies in an infinite sequence of divided powers. Introduction. K. Newman has shown us a counter-example to [3, Theorem 2, p. 521]. The theorem states that if H is a cocommutative Hopf algebra over a perfect field k of characteristic p > 0 and k is the unique simple subcoalgebra of H then a primitive element 1x of Hlies in a sequence of divided powers 1 = Ox, 1x, . . . pn+l lx if and only if 1x has coheight n, for n = 0, 1, .. ., oo. (The sequence should be considered infinite if n = oo.) The proof given in [3, p. 524] correctly shows that the existence of the sequence of divided powers implies that 1x has the desired coheight for all n. The proof there also correctly shows that for finite n if lx has coheight n then the desired sequence of divided powers exists. The error seems to be the assertion K1 H1/(U ker Fi) [3, p. 254, lines 23-24]. Newman's example shows that 1x may have infinite coheight-so lies in arbitrarily long finite sequences of divided powers-and yet 1x lies in no infinite sequence of divided powers. We show here that with the further assumption that the primitives of H are finite dimensional then 1x having infinite coheight implies that 1x lies in an infinite sequence of divided powers. This result is important because it plays a key role in the proof of Jacobson's conjecture in [1]. We explain this in more detail at the end of the present paper. 1. Suppose C is a cocommutative coalgebra with a unique simple subcoalgebra which is one dimensional. We identify this simple subcoalgebra with k and so consider kc C. Then the primitive elements of C, P(C){c E C I Ac =1 0 c + c 0 1}. HEYNEMAN'S THEOREM. If dim P(C) < oo then C satisfies the minimum condition and descending chain condition for subcoalgebras. Proof. For a vector space U the coalgebra Sh(U) is defined in [4, p. 244, p. 254]. By [4, p. 254, 12.1.1 ] there is a finite-dimensional space U and an injective coalgebra Received by the editors April 7, 1970. AMS 1969 subject classifications. Primary 1452; Secondary 1450.