Oystein Ore

On a special class of polynomials

In the present paper one will find a discussion of the main properties of a special type of polynomials, which I have called p-polynomials. They permit several applications to number theory and to the theory of higher congruences as I intend to show in a later paper, and they also possess several properties which are of interest in themselves. The p-polynomials are defined in a field with prime characteristic p (modular fields); they form a (usually non-commutative) ring, where ordinary multiplication is replaced by symbolic multiplication, i.e., substitution of one polynomial into another. The p-polynomials are completely characterized by the property that the roots form a modulus. This modulus has a basis, and one shows consequently that the p-polynomials will have a great number of properties in common with differential and difference equations, such that the theory of p-polynomials gives an algebraic analogue to the theory of linear homogeneous differential equations. One finds that the theorems on the representation of differential polynomials will hold also for p-polynomials; the decomposition in symbolic prime factors is not unique, but the factors in two different representations will be similar in pairs. One can introduce the system of multipliers and the adjoint of a p-polynomial and even the Picard-Vessiot group of rationality; it corresponds in this case to a representation of the ordinary Galois group of the p-polynomial by means of matrices in the finite field (mod p). When this representation is reducible, the p-polynomial is symbolically reducible and conversely. In this paper I have given only the fundamental properties in the theory of p-polynomials; various interesting problems could only be mentioned, while most applications of the theory had to be reserved for another communication. There are a few applications to higher congruences in ?5, chapter 1, giving new proofs for theorems by Moore and Dickson; in ?6 I give a new and simplified proof for the theorem of Dickson on the complete set of invariants for the linear group (mod p). The invariants are, as one will see, the coefficients of a certain p-polynomial, and a slight generalization of the proof of the fundamental theorem on symmetric functions gives the desired result.

Contributions to the theory of finite fields

The present paper contains a number of results in the theory of finite fields or higher congruences. The method may be considered as an appUcation of the theory of /»-polynomials, which I have developed in a recent paperf On a special class of polynomials. In this special case the /»-polynomials form a commutative ring. However, this paper may be read without reference to the former investigations and one may say that the method applied is the representation of the finite field in its group ring. It should be mentioned at this point that a number of the results have direct applications in the theory of algebraic numbers. In chapter 1 the special properties of the /»-polynomials with coefficients in a finite field have been derived and the main results are the theorems that every /»-polynomial has primitive roots and that every /»-modulus is simple. A coroUary is the theorem of Hensel, that every finite field has a basis consisting of conjugate elements. Through the introduction of a symboUc multiplication of elements in a /»-modulus we make every such modulus a ring usuaUy containing divisors of zero. The results of this first chapter I have previously given without proofs.J In chapter 2 various theorems of decomposition and theorems on prime polynomials belonging to a product of /»-polynomials have been derived. Theorems 4 and 5 seem to be the most interesting of the results. In the next chapter these results are applied to the construction of irreducible polynomials. Theorem 1 gives a general type of irreducible polynomials. Next the complete prime polynomial decomposition of the simplest /»-polynomials are given, and it is shown how most known irreducible polynomials (mod /») can be obtained in this way, thus obtaining a unified method for deriving various formerly known results. In the last paragraph one finds a new class of irreducible polynomials closely related to the Unear fractional substitutions. The last chapter contains a few rudiments of the theory of finite fields considered as cyclic fields and also a particularly simple proof for the general law of reciprocity.

Graphs and subgraphs

of the general theorem about the existence of subgraphs with prescribed local degrees. The criterion obtained is related to a criterion established by Tutte but it is considerably simpler in its application, through the fact that it refers only to the properties of single subsets of the vertex set, while the criterion of Tutte involves the choice of pairs of subsets. The proof is based on the alternating path method originally introduced by Petersen in graph theory, and subsequently used by most writers studying the existence of subgraphs, let us mention only the more recent papers by Baebler, Beick, Gallai and Tutte. Due to the applications our presentation of the alternating path theory differs in certain respects from the previous ones, but to save space the proofs have been based, as far as possible, upon those given by the preceding authors. The references are made to the paper by Tutte [12] which should be readily available to most readers. In Chapter 3 one finds various new explicit factorizations for nonregular graphs of certain types. It is pointed out how all the known results on the factorization of regular graphs, in particular those by Baebler, Gallai and Tutte, follow as special cases. To conclude one finds an example to show that a certain limit for the factorization of odd regular graphs given by Baebler is

On the theorem of Jordan-Hölder

In a group we have the well known theorem on principal series, that any two principal maximal series have the same length and the quotient groups in the two series are isomorphic in some order. In a paper entitled Über die von drei Moduln erzeugte Dualgruppe, Dedekindf analyzed the axiomatic foundation of this theorem, particularly the fact that the length of two maximal principal series is the same. He showed that this theorem can be considered as a theorem on structures (Dualgruppen), i.e., systems with two operations called union and cross-cut. In order that the theorem shall hold in such a structure it is necessary that it satisfy a further condition which I have caUed the Dedekind axiom. Similar considerations have been made by G. Birkhoff.J In a recent paper § I have shown that in Dedekind structures one can prove a general theorem corresponding to the theorem of SchreierZassenhaus|| for principal series in groups. This theorem contains the analogue of the theorem of Jordan-Holder for Dedekind structures and yields also the fact that the quotients are isomorphic in some order. All these investigations apply only to Dedekind structures, and give the analogues to the theorems on principal series, i.e., series of sub-groups where each group is a normal sub-group of the full group. They do not apply to composition series where one only supposes that each term is normal under the preceding. In this paper we shall investigate the possibility of deriving a theory applicable to arbitrary structures and giving an analogue to the theorem of Jordan-Holder for composition series. The first step is to examine the validity of the analogue to the second theorem of isomorphism (Theorem 1). Next we have to introduce some notion of normality and normal elements. It turns out that two suitable types of normality, aand /3-normality may be

Graphs and subgraphs. II

title. The terminology is unchanged and the enumeration from the first paper is continued. We suppose as before that the basic graph G is finite and without loops. We observe that by small reformulations in certain statements loops could have been included in the theory, while it is essential for several results that G be finite. Our starting point in Chapter 4 is the theorem of Petersen about the interrelation between conformal subgraphs (subgraphs with the same local degrees). The choice available in the determination of the edges in the desired subgraph H leads to the concept of free equivalence as well as to a unique decomposition of the graph into a bound and a free part. Criteria are established to determine when an edge is free or bound. These are applied, in particular, to the subgraphs with constant proportions for the local degrees. The existence of such subgraphs was established in Chapter 3. Here it is shown that for these all edges are free equivalent; hence the same is true for the regular graphs and subgraphs discussed in ?3.2. A special case is a well known result by Petersen for subgraphs of first degree in regular graphs of degree 3 without peninsulas. It is of interest to note that this particular theorem has an important application for the method of alternating paths in general graph theory. In ?4.4 it is shown that the accessible characters of vertices under alternating H-paths is invariant, that is, do not depend on H but only upon the class of conformal subgraphs to which H belongs. In Chapter 5 the concept of free equivalence is discussed in greater detail. Its relation to the so-called cursal equivalence is examined. Among the results are criteria for two vertices to have the same accessible set with identical cursal properties. There exist a considerable number of problems related to those analysed, but these may be left to others. Chapter 6 contains observations on regular graphs which are completely decomposable, that is, are the sum of subgraphs of first degree.