# Olive Clio Hazlett's papers

Below we list the sixteen papers that Olive C Hazlett published between 1914 and 1930. These are technical works and so, even the excellent historical introductions to these papers require one to have a deep understanding of the mathematical ideas involved. We give extracts from the papers, mostly the first introductory section, for those with the necessary mathematical background.

1. Olive C Hazlett, Invariantive Characterization of Some Linear Associative Algebras, Annals of Mathematics (2) 16 (1/4) (1914-1915), 1-6.

Linear associative algebras of a small number of units, with coordinates in the field of ordinary complex numbers, have been completely tabulated and their multiplication tables have been reduced to very simple forms. But if we had before us a linear associative algebra, the chances are that its multiplication table would not be in such a form that we could find out readily to what standard form it was equivalent. And so the question arises, "May we not find invariants which completely characterize these algebras?"

We use the term invariant here in the sense defined by Professor Dickson, - that is, a single-valued function (in Dirichlet's sense) of the constants of multiplication which takes on the same value for two algebras of the same number of units (having constants of multiplication and coordinates in any field $F$) if the two algebras be equivalent with respect to F, - that is, if there be a non-singular linear homogeneous transformations with coefficients in F which will transform the units of one algebra into the units of the second algebra. The invariants $I_{1}, ..., I_{m}$ are said to completely characterize the algebras of $n$ units over a field $F$ when each $I_{k}$ has the same value for two algebras only when they are equivalent.

It turns out that, in the field $\mathbb{C}$ of ordinary complex numbers, under the general group $T$ of linear transformations, the binary and ternary linear associative algebras with a modulus are both completely characterized by the rank of the algebra and by the rank of certain homogeneous quadratics in the constants of multiplication and the coordinates of the general number of the algebra.

In order to find invariants in the more technical sense defined above, we will first find some rational functions which, in the more usual sense, remain invariant under any linear transformation on the units. We will mean by a rational invariant of the $f$-ary linear algebra $(e_{1}, ..., e_{n})$ with coordinates in $F$ and indeterminate constants of multiplication $\gamma_{ijk} (i, j, k = 1, ..., n)$, a rational function of the $\gamma$'s which, when the units of the algebra are subjected to any linear transformation with coefficients in $F$, is merely multiplied by some power of the determinant of the transformation. Rational covariants are defined in a similar manner.

2. Olive C Hazlett, On the Rational, Integral Invariants of Nilpotent Algebras, Annals of Mathematics (2) 18 ( 2) (1916), 81-98.

In the study of linear algebras, their invariantive classification is one of the most important problems. Peirce, Study, Scheffers, Hawkes, and others have classified, over the field C of ordinary complex numbers, linear associative algebras in a small number of units; and associative algebras having less than four units, with a modulus, are easily characterized invariantively by invariants obtainable from the characteristic determinants, but such invariants are not sufficient for the characterization of quaternary associative algebras with a modulus. In view of Wedderburn's generalization of a theorem of Cartan, we can go a long way toward characterizing the general linear algebra if we can characterize three special kinds of algebras; namely, matric algebras, division algebras, and nilpotent algebras. Over the field $\mathbb{C}$, there is only a finite number of classes of the first two kinds of algebras of a given order; whereas, over the field $\mathbb{C}$, there is an infinite number of classes of nilpotent algebras of any order greater than two; and hence, we centre our attention on nilpotent algebras. In this paper, we consider rational, integral invariants of such algebras. Now all invariants obtainable from the characteristic determinants are zero for nilpotent algebras; and furthermore, there are no rational, integral invariants (other than constants) for $n$-ary nilpotent algebras under the total $n$-ary linear group. We accordingly consider invariants under the group which leaves unaltered the canonical form. For such rational, integral invariants this paper proves theorems analogous to theorems about invariants of algebraic forms, and in particular proves their finiteness. The fundamental invariants are found for the simpler cases.

3. Olive C Hazlett, On the Theory of Associative Division Algebras, Transactions of the American Mathematical Society 18 (2) (1917), 167-176.

There is a famous theorem to the effect that the only linear associative algebras over the field of all real numbers in which division is uniquely possible are the field of real numbers, the field of ordinary complex numbers, and real quaternions. The first published proof of this was that given in 1878 by Frobenius in his fundamental memoir on bilinear forms. Since this proof, there have been numerous others, the most recent being one by Professor Dickson. In the last mentioned proof, the theorem is a special case of a more general theorem of the same nature for a certain class of algebras, which Dickson calls Type A. He defines an algebra of this type as a linear associative algebra $A$, the coordinates of whose numbers range over any given algebraic field F, and for which the following properties hold:

(a) There exists in $A$ a number $i$ satisfying an equation $\phi(x) = 0$ of degree $n$ with coefficients in $F$ and irreducible in $F$.

(b) Any number of $A$ which is commutative with $i$ is in $F(i)$.

(c) There exists in $A$ a number $j$, not in $F(i)$, such that $ji = \theta j + \sigma$, where $\theta$ and $\sigma$ are in $F(i)$.

All three of these conditions are satisfied by real quaternions, and the first two by any linear associative division algebra $D$ over $F$, where we take $i$ so that the degree of the irreducible equation in $F$ satisfied by $i$ is the maximum.
...
The present paper considers linear associative division algebras over a general algebraic field $F$, which may be described as sets of numbers satisfying all the conditions for a field, except that multiplication is not necessarily commutative. It turns out that a necessary and sufficient condition that such an algebra satisfy (c) is that 0 be a root of a certain algebraic equation .... Then, from some fundamental properties of the equation, we show, among other theorems, that, if a linear associative division algebra of a certain general type over an algebraic field $F$ be of rank $n$, it is of order $mn$ where $m ≤ n$. Of this theorem, Frobenius's theorem about real quaternions is a corollary.

4. Olive C Hazlett, On Scalar and Vector Covariants of Linear Algebras, Transactions of the American Mathematical Society 19 (4) (1918), 408-420.

This paper concerns itself with two rather different kinds of covariants of the general linear algebra, which might, for convenience, be distinguished by the adjectives scalar and vector. Consider the general linear algebra $E$ with the units $e_{1}, ..., e_{n}$ and with the constants of multiplication $\gamma_{ijk} (i, j, k = 1, ..., n)$, where the general number of the algebra is $X = \Sigma x_{i} e_{i}$. In a previous paper we defined a rational integral covariant $C$ of the algebra $E$ as a rational integral function of the $y$'s and the $x$'s which possesses the invariantive property whenever the units are subjected to a linear transformation. The paper just mentioned uses such covariants to characterize linear associative algebras in two and three units. The present paper proves for such covariants several fundamental theorems analogous to the fundamental theorems in the theory of invariants for algebraic forms. Since these covariants are isobaric and homogeneous, we can apply here Hilbert's proof of the "finiteness" of the number of covariants of algebraic forms.

We must not, however, content ourselves with the study of such covariants, since they are not sufficient to characterize linear algebras, both associative and non-associative, even when there are only two units. Accordingly, we consider rational integral functions of the $y$'s, the $x$'s and also the units $e_{i} (i = 1, ..., n)$ which possess the invariantive property. Since these functions involve the $e$'s, we call them vector covariants in contradistinction to the functions $C$ mentioned above, which we might call scalar covariants. At first sight, it might seem as if we were confronted with a rather inconvenient difficulty for the following reason. In a theory of such covariants we would expect to find treated certain differential operators analogous to the familiar annihilators $\Omega$ and 0; but Scheffers has shown that, in a linear algebra, a derivative is not uniquely determined unless multiplication is commutative, and we are concerned with algebras both commutative and non-commutative. This Gordian knot can, however, be readily cut by a device. We can, accordingly, derive the annihilators in the approved manner, and hence show that every rational integral vector covariant of the linear algebra is a covariant of the general number of the algebra $X = \Sigma x_{i} e_{i}$ and a suitable set of scalar covariants of the algebra. From this fact flow theorems analogous to those proved for scalar covariants and, in particular, the "finiteness" of vector covariants.

5. Olive C Hazlett, A Theorem on Modular Covariants, Transactions of the American Mathematical Society 21 (2) (1920), 247-254.

This paper answers a question which was raised over five years ago, but which has not been answered so far as I know. Miss Sanderson's theorem on the relation between formal and modular invariants for the Galois Field $GF [p^{n}]$ of order $p^{n}$ enabled her to construct covariants of a system $S$ of binary forms in $x$ and $y$ from invariants of this system $S$ and an additional linear form. This is closely analogous to the situation in the theory of algebraic invariants. In the latter theory the converse also is known to be true that is, we can form all covariants in this manner. In the case of modular invariants, however, we do not obtain all covariants in this way, for the universal covariant $L = x^{p^{n}} y - xy^{p^{n}}$ can not be obtained as a modular invariant of a linear form, since it vanishes whenever $x$ and $y$ are in the field $GF [p^{n}]$, as we suppose the coefficients of our forms to be. In the paper referred to above, Miss Sanderson raised the question as to whether all covariants of a system $S$ can be expressed as polynomials in $L$ and the modular invariants of the system $S$ enlarged by a linear form. The present paper answers this question in the affirmative.

6. Olive C Hazlett, New Proofs of Certain Finiteness Theorems in the Theory of Modular Covariants, Transactions of the American Mathematical Society 22 (2) (1921), 144-157.

In a recent paper of mine it was proved that every modular covariant of a system of forms $S$ (with variables $x$ and $y$) is a polynomial in the universal covariant $L$ and modular invariants of the system of forms $S$ (with variables $\xi$ and $\eta$) enlarged by the linear form $\eta x - \xi y$ which have been made formally invariant as to $x$ and $y$. As pointed out in that paper, we have as a corollary the following:
If $K$ is the class of all modular concomitants of the system $S$ which are formally invariant as to certain sets of coefficients and variables, but not formally invariant as to $x$ and $y$, then the theorem tells us how to construct the set $K'$ of all modular concomitants which are formally invariant as to $x$ and $y$ in addition to being formally invariant as to those sets of coefficients and variables with respect to which $K$ is formally invariant.
Here $x$ and $y$ may be the variables of the system $S$ or a pair of variables which is cogredient with the variables of the system or even a pair of variables which is cogredient with the variables aside from a power of the determinant of the transformation. In fact, every modular covariant of the set $K'$ is a polynomial in $L$ and the concomitants of the set $K$ which have been made formally invariant as to $x$ and $y$. In the present paper we give a few extensions and applications of this theorem.

7. Olive C Hazlett, Associated Forms in the General Theory of Modular Covariants, American Journal of Mathematics 43 (3) (1921), 189-198.

In any theory of covariants, it is of prime importance to ascertain whether or not all covariants of the set are expressible as functions of the covariants belonging to a finite subset. We may attack this fundamental problem from either one of two different points of view: either we may endeavour to express all covariants of the set as rational integral functions of the covariants of a finite subset, or we may content ourselves with the problem of finding rational relations (syzygies) connecting the covariants. The first leads to the finiteness theorem; the second, to the theory of associated forms. Whichever problem we attack, there emerge two entirely different theories, according as the coefficients of the transformations of the group are marks of a field of characteristic zero or marks of a field of characteristic $p ≠ 0$.

In the theory of algebraic covariants of a system of forms under a group of transformations having the coefficients in a field of characteristic zero, both problems have been successfully attacked. The most important names to be associated with the first problem are, Gordan, Mertens, and Hilbert; with the second problem, we associate the names of Boole, Hermite and Clebsch.

In the theory of algebraic covariants of a system of forms under a group of linear transformations whose coefficients are marks of a field of character $p ≠ 0$, comparatively little has as yet been accomplished on either of these problems. It must be remembered that, in the case of a finite field, there present themselves two distinct kinds of algebraic covariants in contrast to the single kind of algebraic covariant that arises when the field is of characteristic zero. For, in the latter field, if a function be unaltered in form, it is unaltered in value and conversely. Whereas, in a finite field, if a function be unaltered in form it is unaltered in value, but the converse is not true in general, in view of Galois' generalization of Fermat's theorem. Accordingly, then, we distinguish two kinds of algebraic covariants of a system of forms under a group of linear transformations with coefficients in the Galois field $GF[p^{n}]$ - formal (modular) covariants and modular covariants, according as the coefficients of the original form are regarded as independent variables or as marks of the field.

The first to consider formal invariants was Hurwitz, who found that they arose naturally in an inquiry into the number of roots of the congruence $a_{r}x^{r} + a_{r-1}x^{r-1} + ... + a_{1}x + a_{0} \equiv 0 (\mod p)$. He proved the finiteness theorem for formal invariants for the special case where the order of the group G is not divisible by $p$. This case is of only minor importance; for the total linear group of transformations whose coefficients are in the field $GF[p^{n}]$ is of order $p^{n}(p^{n} - 1) (p^{2n} - 1)$ which is congruent to zero modulo $p$. Four years later, Dickson introduced the notion of modular invariants and published an elegant theory of modular invariants in which he proved that there is only a finite number of modular invariants of any system of forms under any group G of linear transformations. Four years later, Dickson proved that the set of all modular covariants of any system of forms possesses the finiteness property, i.e., they are all expressible as polynomials in the covariants belonging to a finite subset. In 1914, one of Dickson's students extended this theorem to the modular invariants of a system of forms and a number of cogredient binary points. Up to the present, the finiteness theorem has been proved for formal covariants only in very special cases - these are due to Professor Glenn. As Hurwitz pointed out, this is a most difficult problem, for none of the methods that obtain in the classical theory of algebraic covariants will apply here.

Hence, since this problem is so intractable, it may be of interest to consider the related problem of syzygies. The present paper extends the method and results of Hermite's fundamental memoir on associated forms for ordinary algebraic covariants to modular covariants, both formal and otherwise. The main theorem proves that, if $f = a_{0}x^{m} + a_{1}x^{m-l} + ...$ is a binary form of order not divisible by $p$, then any modular covariant of $f$ for the Galois Field $GF[p^{n}]$ of order $p^{n}$ is expressible (aside from a power of $f$) as a polynomial in the universal covariants $Q$ and $L$, where the coefficients of the terms in $Q$ are polynomials in the forms associated with $f$. We also prove an analogous theorem for formal covariants. From the first of these we prove the rather striking corollary that, aside from a power of $f$, every modular covariant is congruent to an ordinary algebraic covariant of $f$ whenever the variables $x$ and $y$ are in the field. Similarly we prove that, aside from a power of $a_{0}$, every modular invariant is congruent to an ordinary algebraic invariant of $f$. Another corollary gives a neat method of constructing a modular covariant having a given leader, provided the leader has $a_{0}$ as a factor. The main theorem, together with its corollaries, is verified for the binary quadratic, modulo 3.

8. Olive C Hazlett, A Symbolic Theory of Formal Modular Covariants, Transactions of the American Mathematical Society 24 (4) (1922), 286-311.

Thus far, very little has been published on the general theory of formal modular invariants or covariants. Workers have, on the whole, obtained results for special, more or less isolated, cases; and although some beautiful and important general theorems have been proved, they are more or less unrelated. This is, of course, only natural in any division of knowledge in its formative state.

Nevertheless, no worker in the field could fail to be conscious of a certain uniformity common to the special cases that have been studied in detail; though (alas!) this uniformity usually appeared to be broken ruthlessly in the next case studied. This breaking of an apparent law signified, however, merely that we did not know these special cases with a sufficient thoroughness of illuminating detail, or were trying unwittingly to make the laws conform to certain standards, unconsciously preconceived. This latter handicap was laid on us naturally enough by our thorough knowledge of algebraic invariants and the fact that this newer kind of covariants is, in many ways, strikingly like the older, classic covariants, though so tantalisingly different.

Their similarity and their difference show themselves in the very beginning of the study: in the definitions, in the simplest examples. Perhaps the differences that first come to mind are those which are inherent in the fields of definition, which, in the case of classic covariants, is the field of reals or ordinary complex numbers and, in the case of modular covariants, is a Galois field, $GF[p^{n}]$, of order $p^{n}$. These differences are too obvious to mention in detail, but one who has studied the beautiful proofs given by the old masters of invariant theory has been forced to the conclusion that most of the proofs seemed to use the properties of a field of characteristic zero, not in some accidental manner, but rather in veriest necessity.

Growing from the surface differences between the two fields are two very important distinguishing characteristics of the two kinds of covariants. It is well known that an algebraic covariant is necessarily such that all of its terms have the same weight - i.e., it is isobaric; but a modular covariant is not necessarily isobaric. Nevertheless, although its terms are not in general of exactly the same weight, their weights can differ at most by multiples of $p^{n} - 1$. Moreover, if two classic invariants are identical in value for all marks of the field, they are necessarily identical in form, and conversely; whereas, if two formal modular invariants are identical for all marks of the field of definition, they are not necessarily identical in form. Nevertheless, although two such invariants can be of different degree and appear quite different in form at a casual glance, yet they necessarily have in common certain fundamental characteristics. There are other differences that will occur to any worker in the field, but I think that I have mentioned the most refractory.

As indicated above, in spite of the great differences between the algebraic covariants and formal modular covariants, there are certain fundamental likenesses which are more easily sensed than they are analyzed. A feeling that there is some theory which underlies all the special cases, and yet which is comparatively simple, made the writer try to crystallize this theory into words.

In the spring of 1918, came the feeling that the theory of formal modular covariants of a binary form, $f$, for the field $GF[p^{n}]$, must be, in essentials, equivalent to the theory of simultaneous algebraic covariants of f and certain other forms obtained from $f$ by replacing the coefficients of $f$ by their respective $p^{n}$th powers. This is natural enough, since these powers of the coefficients of $f$ are cogredient with the coefficients of $f$ for the transformations of the group used, and this is the only way in which a formal modular covariant differs from an algebraic covariant of $f$. Then there appeared other indications that there is an intimate relation between formal modular covariants of $f$ and algebraic covariants of a system of forms consisting of $f$ and related forms. This paper is an attempt to put in systematic form the theorems which emerged when these eventually crystallized.

9. Olive C Hazlett, Annihilators of Modular Invariants and Covariants, Annals of Mathematics (2) 23 (3) (1922), 198-211.

Up to the present, not very much has been accomplished toward developing a theory of modular covariants for the general case - i.e., for the general form or for the general field. Dickson has proved that modular invariants and covariants of any system of binary forms possess the finiteness property when the coefficients of the transformations are marks of any Galois Field $GF[p^{n}]$ of order $p^{n}$, and fundamental sets of invariants, seminvariants and covariants have been found by various writers for the more important special cases. But in these latter papers, one is struck by the fact that the methods used are ones which apply admirably to the cases considered but in some way fail of complete generality.

Nevertheless, the results for the different special cases have analogies, some of which are rather striking. Some of these analogies are shown in the conditions that a given function so be a seminvariant of a form $f$ and in the closely related subject of annihilators of invariants and covariants. A few years ago, Professor Dickson found annihilators of modular seminvariants of the binary quadratic and binary cubic analogous (in a general way) to those in classical invariant theory. Then he found a set of annihilators for modular seminvariants of a binary quadratic (or cubic) for some of the Galois Fields $GF[p^{n}]$ of order $p^{n}$, where $p$ is a small prime and $n$ is greater than 1. This gives at once necessary conditions that a polynomial $\phi$ be a seminvariant of $f$. For the cases considered, he verified that these conditions are also sufficient.

The present paper attacks the problem in a slightly different way and obtains results which apply to any system of binary forms and any Galois Field $GF[p^{n}]$ of order $p^{n}$. The annihilators of modular invariants obtained in this manner are of the type anticipated in the paper by Professor Dickson; so also are the set of necessary and sufficient conditions that a polynomial $\phi$ be a modular seminvariant. It is interesting to note that these operators are also annihilators of formal modular invariants if no reductions are made by Galois' generalization of Fermat's Theorem $(a^{p^{n'}} \equiv a)$. Hence, since the modular covariants of a system $S$ may be obtained from the modular invariants of an enlarged system $S'$, we readily have annihilators of modular covariants, In the same manner, we obtain annihilators of formal modular covariants. These annihilators lead to a set of $n$ necessary and sufficient conditions that a polynomial $\phi$ be a modular covariant (formal or otherwise).

10. Olive C Hazlett, On the arithmetic of a general associative algebra, Proceedings of the International Mathematical Congress 1 (University of Toronto Press, Toronto, 1924), 185-191.

After having sent my abstract to the Secretary of the Congress, I discovered that Professor Dickson was presenting to the Congress a paper in which he proved the same results as those obtained in my paper. My work, however, was done quite independently; in fact, my rough draft is dated May 30, 1924. The coincidence of our results does not seem surprising in view of the fact that they seem the natural and inevitable ones after those obtained in Dickson's admirable book, Algebras and their Arithmetics.

In his most recent book, Dickson defined an integer for an associative algebra, having a modulus designated by 1, over the field of rational numbers, and from this definition has developed a beautiful theory of such integers. Unlike the earlier definitions of Lipschitz, Hurwitz and Du Pasquier, this definition leads to unique factorization into primes for the classic algebras in two and three units. There he says that each element of a set of elements of such an algebra $A$ shall be called an integer if the set has the following four properties:
R: For every element of the set, the coefficients of the rank equation are all integers.

C: The set is closed under addition, subtraction and multiplication.

U: The set contains the modulus 1.

M: The set is a maximal (i.e., it is not contained in a larger set having properties R, C and U).
Dickson showed that, if an algebra is an algebraic field, its unique maximal set of integers (according to the above definition) is composed of all the integral algebraic numbers of the field. Moreover, he proved that the study of integers for the general associative algebra over the field $\mathbb{Q}$ of rational numbers reduces to the study of integers for division algebras over the field $\mathbb{Q}$.

After the aforementioned work on integers for algebras over the field of rational numbers, he proves that the arithmetic of any linear associative algebra the coordinates of whose numbers range over all complex numbers is associated with the arithmetic of a direct sum of algebras each with a single unit, and that any number whose norm is not zero decomposes into primes uniquely. Although I agree thoroughly with the results, yet it seems to me that the proofs (insofar as they involve the notion of a set of integers for a complex algebra) are open to objection on purely aesthetic grounds.

His definition of integers of an algebra $A$ over the field of complex numbers does not seem to be the one which is most natural or most artistic. For, since an integer of an algebra over the field of rational numbers is said to have the property R when the coefficients of the rank equation are rational integers, then when we are dealing with an algebra over any field $F$, it seems only consistent so to define an integer of this algebra that it shall be said to have the property R when the coefficients of the rank equation are integers of this same field, $F$. Although he formulates a definition of a set of integers only for the case when the algebra is over the field of rational numbers, yet he appears to use the same definition for the algebras over the field of complex numbers as evidenced by his remarks near the beginning of the discussion in each case.

For example, let the algebra $A$ be the totality $\mathbb{C}$ of all ordinary complex numbers of the form $x + iy$ when $x$ and $y$ are rational. Then we can regard $A$ as an algebra of two units, 1 and $i$, over the field of rational numbers or as an algebra of one unit, 1, over the field of complex numbers of the above form $x + iy$. If we use a definition analogous to Dickson's and hence require that the coefficients of the rank equation be rational integers in each case, then we discover that, in the first case, we get the familiar complex integers; whereas, in the second case, we get merely the rational integers. But, for aesthetic reasons, we should consider it a desideratum so to define integers for the field $\mathbb{C}$ that we get exactly the same set of integers, whether we regard $\mathbb{C}$ as an algebra of two units over the field of rational numbers or as an algebra of one unit over the field $\mathbb{C}$ itself.

In view of these two reasons, we propose the following
DEFINITION. Let $A$ be an algebra over any field $F$. Then each element of a set of elements of $A$ shall be called an integer of $A$ over $F$ if the set has the properties R*, C, U and M, where R* is defined thus, -

R*: For every element of the set, the coefficients of the rank equation are algebraic integers of the field $F$.
If $A$ is any algebra of order $n$ over an algebraic field $F$ of order $k$, then the algebra $A$ can be regarded as an algebra, $A^{*}$, of order $nk$ over the field $\mathbb{Q}$ of rational numbers. We readily prove that a set of integers of $A$ over $F$ possessing properties R*, C and U coincides with a set of integers of $A^{*}$ over $\mathbb{Q}$ possessing properties R, C and U; and conversely. Furthermore, it then follows that the study of integers for the general associative algebra over the field $F$ reduces to the study of integers for division algebras over the field $F$. Moreover, since a division algebra over a field $F$ is equivalent, in a suitable extension of $F$, to a simple matric algebra and since Dickson has shown that a simple matric algebra has a maximal set of integral elements, it follows readily that a division algebra over $F$ has a maximal set of integral elements. Accordingly, any associative algebra over $F$ has a maximal set of integral elements.

11. Olive C Hazlett, The Arithmetic of a General Algebra, Annals of Mathematics (2) 28 (1/4) (1926), 92-102.

This paper concerns itself with a general associative algebra over any algebraic field $F$ and in particular with its arithmetic. In many ways it is merely an enlargement and extension of a paper read by the writer before the Toronto Congress. It uses the definition of an arithmetic of such an algebra which was given by the writer in the paper just mentioned and which replaces the property R of Dickson's definitions by R*. Now, if $A$ is an algebra over an algebraic field $F$, it can be regarded as an algebra $A^{*}$ over the field $\mathbb{Q}$ of rational numbers. In fact, a set of numbers of $A$ over $F$ which has properties R*, C, U, X coincides with a set of numbers of $A^{*}$ over $A$, which has properties R, C, U, M.

It is well-known that an algebra $A$ over $F$ is expressible as the sum of two algebras, $S$ and $N$, where $S$ is a semi-simple subalgebra and $N$ is the maximal invariant nilpotent subalgebra of $A$, and it is here proved that the arithmetic of $A$ is associated with the arithmetic of $S$. This theorem was first proved by Dickson in his beautiful book on Algebras and their arithmetics, but it was proved only for algebras over the field $\mathbb{Q}$ of rational numbers and, moreover, used the normalized units of Cartan and the notion of character, whereas the present proof does not use these units or the notion of character and applies to an algebra over any non-modular field.

12. Olive C Hazlett, Notes on Formal Modular Protomorphs, American Journal of Mathematics 49 (2) (1927), 181-188.

In a recent paper, W L G Williams has proved several theorems relating to formal modular protomorphs of a certain class of binary forms with respect to the general Galois field. ... Unfortunately, however, his proofs are such that they hold only for a form or system of forms for which no binomial coefficient is congruent to modulo $p$ and hence one wonders if the theorems are true for the general form or system of forms.

13. Olive C Hazlett, Homogeneous Polynomials with a Multiplication Theorem, Transactions of the American Mathematical Society 31 (2) (1929), 223-232.

If f(x) is any homogeneous polynomial in the $n$ variables, $x_{1}, ..., x_{n}$, such that
(1) $f(x)f(\xi) = f(X)$
where the $X$'s are bilinear functions of the $x$'s and $\xi$'s,
(2) $X_{k} = [ \sum _{i,j} \gamma_{ijk} x_{i} \xi_{j} ]$   $(k = 1, ..., n)$.
then $f(x)$ is said to admit the composition (2).

One of the simplest such functions is $f(x) = x_{1}^{2} + x_{2}^{2}$ for which
(3) $X_{1}= x_1 \xi_{1}± x_{2}\xi_{2}, X_{2}= x_{2}\xi_{1}± x_{1}\xi_{2}$
This was known to Diophantus. Euler and Degen noticed that, similarly, a sum of n squares admits composition when $n$ = 4 or 8, and Hurwitz proved that such is true only when $n = 1, 2, 4, 8$. Two other well known examples are the multiplication theorem for determinants whose $n^{2}$ elements are independent variables and the norms of algebraic numbers. Moreover the composition of forms occurs in number theory. Usually in number theory we are concerned with the composition of forms in a more general sense, viz,
(4) $f'(x)f''(t) = F(X)$
where$f', f'$' and $F$ are polynomials in same degree, but not necessarily with the same coefficients. Dickson? proved that the determination of three polynomials satisfying (4) can be reduced to the determination of one polynomial satisfying (1). Hence we can and shall restrict our attention to compositions (1).

14. Olive C Hazlett, On Division Algebras, Transactions of the American Mathematical Society 32 (4) (1930), 912-925.

Linear algebras have been studied in great detail for many years, but they have usually been studied over a field $F$. In the papers by Dickson, Cecioni, A A Albert and others, a division algebra over a field $F$ can be regarded as a division algebra defined over an algebra containing $F$. In the present paper, it is proved that every division algebra over a field $F$ can be regarded as an algebra $E_{1}$ over a division algebra $D_{1}$, where $D_{1}$ is in its turn an algebra $E_{2}$ over an algebra $D_{2}$, etc. where each of the component algebras $D_{i}$ is a division algebra defined over a field $F_{i}$ contained in $E_{i}$ of one of the three following types:
(1) fields,

(2) normal Dickson algebras,

(3) algebras which contain no Dickson subalgebras.
This paper does not determine whether there actually exist algebras of the third type, but the writer has the feeling that none such exist. This principle is applied to certain solvable algebras and it appears that under suitable conditions a solvable algebra is a direct product of solvable algebras.

Olive C Hazlett, On formal modular invariants, Journal de mathématiques pures et appliquées (9) 9 (1930), 327-332.

This note gives an extension of the principal results contained in a paper [On the formal modular invariants of binary forms] by W L-G Williams in this journal.

15. Olive C Hazlett, Integers as matrices, Atti del Congresso Internazionale dei Matematici Bologna 1928 (Società Tipografica già Compositori, 1930), 57-62.

In view of the current interest of many mathematicians which converges on the study of ideals for an algebraic field and on generalizations thereof, there may be some interest in the following theorems on the arithmetic of a general linear associative algebra. The pioneer work on the definition and theory of an arithmetic of an algebra which is not a field was done by Lipschitz and A Hurwitz; but, as Dickson pointed out, the definitions there used are wholely unsatisfactory for a general algebra. In fact, Dickson satisfactorily defined an arithmetic of an algebra E over the field of rational numbers to be any set of numbers $x$ of $E$ which possesses the following properties:
R, The rank equation of $x$ has all its coefficients rational integers and the coefficient of the highest power of the variable is unity.

C, The set is closed under addition, subtraction and multiplication.

U, The set contains the modulus.

M, The set is a maximal set of all those sets possessing properties R, C and U that is, it is not contained in any other set possessing properties R, C and U.
For the case of an arithmetic of an algebra over any algebraic field, a satisfactory definition was given by the writer. The present paper uses little
technical knowledge of the theory of algebras. For the essentials of this theory necessary to read this paper, the reader is referred to Dickson's Algebras and their arithmetics (especially Chapters I, VII, VIII) or to its German translation under the editorship of Speiser and Fueter.

Let $E$ be an algebra of order $m$ over any field $F$ and suppose that $A$ is any arithmetic of $E$ of order $a$. Let the numbers of a basis of $A$ be chosen as some of the basal numbers $e_{1}, ..., e_{a}$ of the algebra $E$. Then since $A$ has the property C the totality of numbers of $E$ linearly dependent on the $e_{i} (i = l, ..., a)$ form a subalgebra $D$ of $E$ and by property R the constants of multiplication for $D$ are integers and the first matrices of all numbers of $A$ have integral elements. If $a < m$, there are numbers of $E$ linearly independent of these $e_{i}$. Take as a set of basal numbers of $E$ any set of linearly independent numbers of $E$ which includes the $e_{i} (i = l, ..., a)$. The totality $T$ of first matrices of numbers of $E$ whose elements are integers has properties R, C and U and hence the corresponding numbers of $E$ have properties R, C, U. If $T'$ were a maximal of sets of first matrices possessing properties R, C, U, and containing $T$, then the corresponding numbers of $E$ form an arithmetic of $E$ of order $m$. Hence we have
Theorem 1. Every arithmetic of an algebra has the same order as the algebra.

Last Updated December 2021