Celebratio Mathematica

A. A. Albert: A determination of all associative algebras in two, three and four units over a nonmodular field \( \frak{F} \). M.S. Thesis, University of Chicago, 1927. phdthesis

A. A. Albert: “A determination of all normal division algebras in sixteen units,” Trans. Am. Math. Soc. 31 : 2 (1929), pp. 253–260. An abstract was published in Bull. Am. Math. Soc. 34:4 (1928). MR 1501481 JFM 55.0090.04 article

The chief outstanding problem in the theory of linear associative algebras over an infinite field \( F \) is the determination of all division algebras. This problem is equivalent to that of the determination of all normal division algebras, or algebras \( A \) in which the only elements of \( A \) commutative with every element of \( A \) are the quantities of its reference field \( F \).

The order of a normal division algebra is the square of an integer. All normal division algebras in \( 1^2 \), \( 2^2 \) [Dickson 1927, p. 46], and \( 3^2 \) [Wedderburn 1921, p. 132] units have been determined. In this paper all normal division algebras in \( 4^2 \) units, the next case, are determined and shown to be the algebras of Cecioni [1923].

A. A. Albert: “On direct products,” Trans. Am. Math. Soc. 33 : 3 (1931), pp. 690–711. An abstract was published in Bull. Am. Math. Soc. 37:5 (1931). MR 1501610 JFM 57.0159.01 Zbl 0002.24605 article

The principal contribution of this article is a set of theorems on the structure of the direct product of a normal division algebra \( A \) over \( F \) and an algebraic field \( F(\eta) \), with applications of the Galois theory of equations. The theorems are useful new tools for research on normal division algebras. In particular it is shown that it is possible to extend the reference field \( F \) of any normal division algebra \( A \) of order \( p^2 \) over \( F \), \( p \) a prime, such that \[ A^{\prime} = A\times F(\eta) \] is a cyclic normal divison algebra over \( F(\eta) \). A new proof is given of a little known theorem of R. Brauer [1929] which reduces the problem of determinining all normal division algebras of order \( n^2 \) over \( F \) to the case where \( n \) is a power of a prime.

A. A. Albert and H. Hasse: “A determination of all normal division algebras over an algebraic number field,” Trans. Am. Math. Soc. 34 : 3 (1932), pp. 722–726. An abstract was published in Bull. Am. Math. Soc. 38:3 (1932). MR 1501659 Zbl 0005.05003 article

A. A. Albert: “A note on normal division algebras of order sixteen,” Bull. Am. Math. Soc. 38 : 10 (1932), pp. 703–706. An abstract was published in Bull. Am. Math. Soc. 38:7 (1932). MR 1562492 JFM 58.0141.03 Zbl 0005.34202 article

A. A. Albert: “Non-cyclic algebras of degree and exponent four,” Trans. Am. Math. Soc. 35 : 1 (1933), pp. 112–121. MR 1501674 JFM 59.0158.02 Zbl 0006.15103 article

I have recently [1932] proved the existence of non-cyclic normal division algebras. The algebras I constructed are algebras \( A \) of order sixteen (degree four, so that every quantity of \( A \) is contained in some quartic sub-field of \( A \)) containing no cyclic quartic sub-field and hence not of the cyclic (Dickson) type. But each \( A \) is expressible as a direct product of two (cyclic) algebras of degree two (order four). Hence the question of the existence of non-cyclic algebras not direct products of cyclic algebras, and therefore of essentially more complex structures than cyclic algebras, has remained unanswered.

The exponent of a normal division algebra \( A \) is the least integer \( e \) such that \( A^e \) is a total matric algebra. A normal division algebra of degree four has exponent two or four according as it is or is not expressible as a direct product of algebras of degree two. I shall prove here that there exist non-cyclic normal division algebras of degree and exponent four, algebras of a more complex structure than any previously constructed normal division algebras.

A. A. Albert: “Cyclic fields of degree eight,” Trans. Am. Math. Soc. 35 : 4 (1933), pp. 949–964. An abstract was published in Bull. Am. Math. Soc. 39:1 (1933). MR 1501727 JFM 59.0158.03 Zbl 0008.00201 article

Let \( F \) be any non-modular field, \( C \) be an algebraic extension of degree \( n \) of \( F \). Then \( C=F(x) \) is the field of all rational functions with coefficients in \( F \) of a root \( x \) of an equation \( \phi(\omega)=0 \) which has coefficients in \( F \), degree \( n \), and transitive group \( G \) for \( F \).

The problem of the construction of all equations of degree \( n \) and group \( G \) is evidently equivalent to the problem of the construction of all corresponding fields \( C \). Moreover the construction of a set of canonical equations \( \psi(\omega) = 0 \) with the property that every \( C=F(x) \) of degree \( n \) and group \( G \) is equal to an \( F(y) \) defined by a \( \psi(\omega) = 0 \) provides a solution of both problems.

One of the most important problems in the algebraic theory of fields is the construction of all cyclic fields of degree \( n \) over \( F \). This is the case where \( G \) consists of the \( n \) distinct powers \( S^i \) (\( i = 0 \), \( 1,\dots, n-1 \)) of a single substitution \( S \). In this case \( G \) is also the group of all automorphisms of \( C \). Moreover this problem has been reduced to the case \( n = p^e \), \( p \) a prime.

Cyclic fields of degree 2, \( 2^2 \) have been constructed. In the present paper we shall use purely algebraic methods to construct all cyclic fields of degree \( 2^3 = 8 \) [Mertens 1916] over any non-modular field \( F \).

A. A. Albert: “On normal Kummer fields over a non-modular field,” Trans. Am. Math. Soc. 36 : 4 (1934), pp. 885–892. MR 1501774 JFM 60.0915.03 Zbl 0010.14902 article

Let \( F \) be any non-modular field, \( p \) an odd prime, \( \zeta \neq 1 \) a \( p \)th root of unity. Suppose that \( \mu \) in \( F(\zeta) \) is not the \( p- \)th power of any quantity of \( F(\zeta) \) so that the equation \( y^p=\mu \), is irreducible in \( F(\zeta) \). Then the field \( F(y,\zeta) \) is called a Kummer field over \( F \).

In the present paper we shall give a formal construction of all normal Kummer fields over \( F \). This is equivalent to a construction of all fields \( F(x) \) of degree \( p \) over \( F \) such that \( F(x,\zeta) \) is cyclic of degree \( p \) over \( F(\zeta) \). In particular we provide a construction of all cyclic fields of degree \( p \) over \( F \).

We shall also apply the cyclic case to prove that a normal division algebra \( D \) of degree \( p \) over \( F \) is cyclic if and only if \( D \) contains a quantity \( y \) not in \( F \) such that \( y^p=\gamma \) in \( F \).

A. A. Albert: “On the construction of Riemann matrices. I,” Ann. Math. (2) 35 : 1 (January 1934), pp. 1–28. MR 1503140 JFM 60.0908.02 Zbl 0010.00304 article

A. A. Albert: “On a certain algebra of quantum mechanics,” Ann. Math. (2) 35 : 1 (January 1934), pp. 65–73. MR 1503142 JFM 60.0902.03 Zbl 0008.42104 article

P. Jordan, J. von Neumann, and E. Wigner [1934] have discussed certain linear real non-associative algebras of importance in quantum mechanics. Their algebras \( \mathfrak{M} \) satisfy the ordinary postulates for addition, the commutative law for multiplication, and the distributive law, but they are non-associative.

In the paper quoted above it is shown that, with a single exception, every algebra satisfying the above postulates is equivalent to an algebra \( \mathfrak{M} \) whose elements are ordinary real matrices \( x, y,\dots \) with products \( xy \) in \( \mathfrak{M} \) defined by quasi-multiplication, \[ xy = \tfrac{1}{2}(x\cdot y + y\cdot x), \] where \( x\cdot y \) is the ordinary matrix product. This single exception is the algebra \( \mathfrak{M}_3^8 \) of all three rowed Hermitian matrices with elements in the real non-associative algebra \( C \) of Cayley numbers.

The algebras obtained by quasi-multiplication of real matrices were considered in earlier papers so that, as stated by the above authors, this seemingly exceptional case, if proved really exceptional, is the only algebra of the above type which could lead to any new form of quantum mechanics.

In the present paper I shall prove that \( \mathfrak{M}_3^8 \) is a new algebra and that it is not equivalent to any algebra obtained by quasi-multiplication of real matrices. Moreover, I shall show that the relation \[ x(yx^2) = (xy)x^2 \] is satisfied by \( \mathfrak{M}_3^8 \).

A. A. Albert: “A solution of the principal problem in the theory of Riemann matrices,” Ann. Math. (2) 35 : 3 (July 1934), pp. 500–515. An abstract was published in Bull. Am. Math. Soc. 40:3. MR 1503176 Zbl 0010.00401 article

The principal problem in the theory of Riemann matrices is that of determining the multiplication algebra \( D \) of any pure Riemann matrix \( \omega \). This problem is of great importance in the transcendental theory of algebraic geometry, and its solution provides a solution of many related algebraic geometric questions from which it arose.

Algebra \( D \) is a normal division algebra of degree \( n \) (order \( n^2 \)) over its centrum \( K \). The centrum \( K \) is an algebraic field \( R(\rho) \) of all rational functions with rational coefficients of an abstract root \( \rho \) of an irreducible rational equation \( f(r) = 0 \). We call \( \omega \) a Riemann matrix of the first or second kind according as \( f(r) = 0 \) does or does not have all real roots.

If \( \omega \) is of the first kind, then [Albert 1932] \( n=1,2 \); I have determined [1934] the resulting algebras \( D \), and have proved [1934] that there exist pure Riemann matrices \( \omega \) with any one of the so determined algebras \( D \) as mutiplication algebra. There remains the case of matrices \( \omega \) of the second kind.

The remaining more difficult problem will be solved in the present paper. I shall first define a type of algebra \( \mathfrak{A} \), shall prove the existence of Riemann matrices \( \omega \) with \( \mathfrak{A} \) as multiplication algebra, and shall finally show that the multiplication algebra \( D \) of any pure Riemann matrix \( \omega \) of the second kind is an algebra \( \mathfrak{A} \).

A. A. Albert: “Normal division algebras of degree 4 over \( F \) of characteristic 2,” Am. J. Math. 56 : 1–4 (1934), pp. 75–86. MR 1507931 JFM 60.0105.01 Zbl 0008.24202 article

I have determined [1932a] all normal division algebras of degree 4 over any non-modular field \( F \) and it is evident that the determination is valid when \( F \) is any infinite modular field of characteristic \( p\neq 2 \). There remains the case \( p=2 \).

In the non-modular case there are but two types of algebras, the cyclic algebras and a second type containing algebras which may or may not be cyclic. I have proved [1932b; 1933] the existence of both primary and non-primary non-cyclic algebras over a non-modular \( F \).

In the present paper I shall assume that \( F \) is any infinite field of characteristic \( p=2 \) and shall determine all normal division algebras \( D \) of degree 4 (order 16) over \( F \). As in the non-modular case the above two types of algebras appear. However I shall prove that every non-primary \( D \) is cyclic. I shall also prove that a necessary and sufficient condition that \( D \) be cyclic is that \( D \) contain a quantity \( t \) not in \( F \) such that \( t^2 \) is in \( F \).

Finally I shall give a construction of the non-cyclic type of algebra \( D \) and shall prove that \( D \) is a non-cyclic primary normal division algebra if and only if a certain quadratic form in nine variables (and with coefficients determined by the multiplication table of \( D \)) is not a zero form.

A. A. Albert: “Integral domains of rational generalized quaternion algebras,” Bull. Am. Math. Soc. 40 : 2 (1934), pp. 164–176. An abstract was published in Bull. Am. Math. Soc. 39:7 (1933). MR 1562813 Zbl 0008.29301 article

A. A. Albert: “Cyclic fields of degree \( p^n \) over \( F \) of characteristic \( p \),” Bull. Am. Math. Soc. 40 : 8 (1934), pp. 625–631. An abstract was published in Bull. Am. Math. Soc. 40:3 (1934). MR 1562919 Zbl 0010.00402 article

A. A. Albert: “On cyclic fields,” Trans. Am. Math. Soc. 37 : 3 (1935), pp. 454–462. MR 1501797 JFM 61.0124.02 Zbl 0011.29005 article

The most interesting algebraic extensions of an arbitrary field \( F \) are the cyclic extension fields \( Z \) of degree \( n \) over \( F \). I have recently given constructions of such fields for the case \( n = p \) [1934a], a prime, when the characteristic of \( F \) is not \( p \), and for the case \( n = p^e \) [1934b] when the characteristic of \( F \) is \( p \). Moreover it is well known that when \( F \) contains all the \( n \)th roots of unity then \( Z = F(x) \), \( x^n = \alpha \) in \( F \).

The last result above does not provide a construction of all cyclic fields \( Z \) over \( F \) since in general \( F \) does not contain these \( n \)th roots. Moreover if we adjoin these roots to \( F \) and so extend \( F \) to a field \( K \) the composite \( (Z,K) \) over \( K \) may not have degree \( n \). Finally even if \( (Z,K) \) over \( K \) does have degree \( n \) then it is necessary to give conditions that a given field \( K(x) \), \( x^n=\alpha \) in \( F \), shall have the form \( (Z,K) \) with \( Z \) cyclic over \( F \). This has not been done and is certainly not as simple as the considerations I shall make here.

It is well known that if \( n = p_1^{e_1}\cdots p_t^{e_t} \) with \( p_i \) distinct primes, then \( Z \) is the direct product \[ Z = Z_1 \times \cdots \times Z_t \] where \( Z_i \) is cyclic of degree \( p \) over \( F \). Hence it suffices to consider the case \( n = p^e \), \( p \) a prime. I have already done so [Albert 1934b] for the case where \( F \) has characteristic \( p \). In the present paper I shall make analogous considerations for the case where \( F \) has characteristic not \( p \) by first studying the case where \( F \) contains a primitive \( p \)th root of unity \( \zeta \) and later giving complete conditions for the case where \( F \) does not contain \( \zeta \).

A. A. Albert: “On the construction of Riemann matrices. II,” Ann. Math. (2) 36 : 2 (April 1935), pp. 376–394. MR 1503230 JFM 61.1037.02 Zbl 0011.38904 article

A. A. Albert: “Involutorial simple algebras and real Riemann matrices,” Ann. Math. (2) 36 : 4 (October 1935), pp. 886–964. See also Proc. Natl. Acad. Sci. U.S.A. 20:12 (1934). MR 1503260 JFM 61.1038.01 Zbl 0012.39102 article

Any algebraic theory depends in part on the properties of its reference field \( \mathfrak{K} \). In particular it is generally desirable to know whether or not the Hilbert Irreducibility theorem holds in \( \mathfrak{K} \).

W. Franz studied [1931] fields with this question in mind and proved that \( \mathfrak{K} \) is a Hilbert Irreducibility field if it is a separable algebraic extension of finite degree over a Hilbert Irreducibility field \( \mathfrak{F} \). This is however insufficient for the theory of algebras over a modular field since the field \( \mathfrak{K} \) may be inseparable over \( \mathfrak{F} \).

We shall treat this latter case here and shall show that any algebraic extension \( \mathfrak{K} \) of finite degree over a Hilbert Irreducibility field \( \mathfrak{F} \) is a Hilbert irreducibility field. Moreover this result will be a sufficiently good tool for the researches of later chapters.

A. A. Albert: “Normal division algebras of degree \( p^e \) over \( F \) of characteristic \( p \),” Trans. Am. Math. Soc. 39 : 1 (January 1936), pp. 183–188. MR 1501840 JFM 62.0101.01 Zbl 0013.10202 article

In a recent paper [1934] I proved that a normal division algebra \( D \) of degree \( p \), a prime, over a field \( F \) of characteristic not \( p \), is cyclic if and only if \( D \) contains a sub-field \( F(y) \), \( y^p = \gamma \) in \( F \). This result evidently leads to the conjecture that any normal division algebra \( D \) of degree \( n \) over \( F \) is cyclic over \( F \) if and only if \( D \) contains a maximal sub-field, \( F(y) \), \( y^n = \gamma \) in \( F \).

The conjectured criterion given above would be of fundamental importance for the theory of the structure of normal division algebras. Without loss of generality we may assume that \( n=p^e \), \( p \) a prime, and the theory then gives rise to two distinct cases according as \( F \) does or does not have characteristic \( p \). We shall consider the former case here and give a brief simple proof of the criterion.

A. A. Albert: “Simple algebras of degree \( p^e \) over a centrum of characteristic \( p \),” Trans. Am. Math. Soc. 40 : 1 (July 1936), pp. 112–126. An abstract was published in Bull. Am. Math. Soc. 41:7 (1935). MR 1501866 JFM 62.0101.02 Zbl 0014.29102 article

The study of normal simple algebras \( A \) over a field \( F \) has been reduced to the case where the degree of \( A \) is a power of a prime \( p \). The theory then splits sharply into two cases distinguished by the hypothesis that the characteristic of \( F \) is or is not \( p \). We shall restrict our attention to the former case.

It is well known that every normal simple algebra \( A \) over \( F \) is similar to a crossed product \( B \). But this result is of little aid in a study of \( A \), since in fact the degree of \( B \) is in general not a power of \( p \). We shall prove here, however, that in our case every \( A \) is similar to a cyclic algebra whose degree is a power of \( p \).

If \( K \) is obtained from \( F \) by adjoining the \( p^e \)th roots of quantities of \( F \) to \( F \), for fixed \( e \), then \( K \) is said to have exponent \( e \) over \( F \). We shall show that the exponent of an algebra \( A \) is \( p^e \) where \( e \) is the exponent of the above \( K \) of least exponent, which splits \( A \). Moreover \( A \) has exponent \( p^e \) only if \( A \) is similar to a direct product of cyclic division algebras \( D_i \) whose exponents and degrees are equal to \( p^{e_i}\leq p^e \), \( D_1 \) of degree \( p^e \).

A. A. Albert: “\( p \)-algebras over a field generated by one indeterminate,” Bull. Am. Math. Soc. 43 : 10 (1937), pp. 733–736. MR 1563627 JFM 63.0875.01 Zbl 0017.24501 article

The structure of all division algebras over the simplest type of non-modular field, the field of all rational numbers, has been determined [Hasse and Albert 1932]. The correspondingly simplest type of infinite modular field is the simple transcendental extension \( K = F(x) \) of a finite field \( F \). Every division algebra \( D \) over such a \( K \) is a normal division algebra of degree \( n \) over a centrum \( G \) which is algebraic of finite degree over \( K \). It is well known that the problem of determining the structure of \( D \) is reducible to the case where \( n \) is a power of a prime \( p \). When \( p \) is the characteristic of \( F \) the algebra \( D \) is called a \( p \)-algebra and we shall solve the problem in this case. Our results will be valid if we replace the finite field \( F \) by any perfect field of characteristic \( p \).

The theorem we shall obtain is remarkable not merely because of the character of the result thus derived but also because of the extremely elementary nature of the proof. By using a simple property of the field \( G \) described above we shall show that every \( p \)-algebra with centrum \( G \) is cyclic and of exponent equal to its degree. Moreover this result is due to the unusual fact that all cyclic algebras over \( G \) of the same degree \( p^e \) have a common pure inseparable splitting field.

A. A. Albert: “On cyclic algebras,” Ann. Math. (2) 39 : 3 (July 1938), pp. 669–682. MR 1503431 JFM 64.0082.02 Zbl 0019.24601 article

A. A. Albert: “Non-cyclic algebras with pure maximal subfields,” Bull. Am. Math. Soc. 44 : 8 (1938), pp. 576–579. MR 1563796 JFM 64.0082.01 Zbl 0019.19302 article

A. A. Albert: Structure of algebras. AMS Colloquium Publications 24. American Mathematical Society (Providence, RI), 1939. MR 0000595 Zbl 0023.19901 book

A. A. Albert: “On ordered algebras,” Bull. Am. Math. Soc. 46 (1940), pp. 521–522. MR 0001972 JFM 66.0114.01 Zbl 0061.05502 article

A. A. Albert: “The radical of a non-associative algebra,” Bull. Am. Math. Soc. 48 (1942), pp. 891–897. MR 0007396 Zbl 0061.05001 article

Every algebra \( \mathfrak{A} \) which is homomorphic to a semi-simple algebra has an ideal \( \mathfrak{R} \). which we shall call the radical of \( \mathfrak{A} \), such that \( \mathfrak{A} - \mathfrak{R} \) is semi-simple, \( \mathfrak{R} \) is contained in every ideal \( \mathfrak{B} \) of \( \mathfrak{A} \) for which \( \mathfrak{A} - \mathfrak{B} \) is semi-simple.

An algebra \( \mathfrak{A} \) is homomorphic to a semi-simple algebra if and only if \( \mathfrak{A} - \mathfrak{A}\mathfrak{H} \) is not a zero algebra.

If \( \mathfrak{A} - \mathfrak{A}\mathfrak{H} \) is a zero algebra and \( \mathfrak{B} \) is an ideal of \( \mathfrak{A} \) the algebra \( \mathfrak{A} - \mathfrak{B} \) is a zero algebra if and only if \( \mathfrak{B} \) contains \( \mathfrak{A}\mathfrak{H} \).

Let \( \mathfrak{A} \) be homomorphic to a semi-simple algebra. Then either \( \mathfrak{A} - \mathfrak{A}\mathfrak{H} \) is semi-simple and \( \mathfrak{A}\mathfrak{H} \) is the radical of \( \mathfrak{A} \) or \( \mathfrak{A} - \mathfrak{A}\mathfrak{H} \) is the direct sum of a semi-simple algebra and a zero algebra \( \mathfrak{R}_0 = \mathfrak{R} - \mathfrak{A}\mathfrak{H} \) such that \( \mathfrak{R} \) is the radical of \( \mathfrak{A} \).

A. A. Albert: “Non-associative algebras. I: Fundamental concepts and isotopy,” Ann. Math. (2) 43 : 4 (October 1942), pp. 685–707. MR 0007747 Zbl 0061.04807 article

A. A. Albert: “Quasigroups. I,” Trans. Am. Math. Soc. 54 (1943), pp. 507–519. MR 0009962 Zbl 0063.00039 article

A. A. Albert: “Two element generation of a separable algebra,” Bull. Am. Math. Soc. 50 (1944), pp. 786–788. MR 0011080 Zbl 0061.05501 article

The minimum rank of an algebra \( A \) over a field \( F \) is defined to be the least number \( r = r(A) \) of elements \( x_1,\dots,x_r \) such that \( A \) is the set of all polynomials in \( x_1,\dots,x_r \) with coefficients in \( F \). In what follows we shall assume that \( A \) is an associative algebra of finite order over an infinite field \( F \).

It is well known that \( r(A) = 1 \) if \( A \) is a separable field over \( F \) and that \( r(A) = 2 \) if \( A \) is a total matric algebra over \( F \). Over fourteen years ago I obtained but did not publish the result that \( r(A) = 2 \) if \( A \) is a central division algebra over \( F \). The purpose of this note is to provide a brief proof of the generalization which states that if \( A \) is any separable algebra over \( F \) then \( r(A) = 1 \) or 2 according as \( A \) is or is not commutative.

A. A. Albert: “On Jordan algebras of linear transformations,” Trans. Am. Math. Soc. 59 (1946), pp. 524–555. MR 0016759 Zbl 0061.05101 article

In the present paper we shall obtain the fundamental structure theorems for Jordan algebras over a nonmodular field. We shall derive analogues for Jordan algebras of the Lie and Engel theorems on solvable Lie algebras and shall obtain a trace criterion for the existence of a nonzero radical. The results imply a Pierce decomposition for a Jordan algebra relative to an idempotent which is almost precisely that known for any associative algebra. Then semisimple Jordan algebras have unity elements and are direct sums of simple algebras. The center of a simple Jordan algebra is a field and every simple Jordan algebra may be expressed as a central simple algebra over its center. If \( \mathfrak{A} \) is central simple every scalar extension \( \mathfrak{A}_{\mathfrak{K}} \) of \( \mathfrak{A} \) is simple.

We shall also show that if \( u \) is any primitive idempotent of a simple algebra \( \mathfrak{A} \) the associative product \( u\mathfrak{A} u \) is a simple subalgebra of \( \mathfrak{A} \) and has \( u \) as unity quantity. We shall call \( \mathfrak{A} \) a reduced algebra if \( \mathfrak{A} \) is simple and \( u\mathfrak{A}u \) has order one for every primitive idempotent \( u \) of \( \mathfrak{A} \). Every central simple algebra has a scalar extension field \( \mathfrak{K} \) such that \( \mathfrak{A}_{\mathfrak{K}} \) is a reduced algebra, and we shall determine all reduced Jordan algebras.

A. A. Albert: “Absolute valued real algebras,” Ann. Math. (2) 48 : 2 (April 1947), pp. 495–501. MR 0020550 Zbl 0029.01001 article

Let \( \mathfrak{D} \) be an algebra over the field \( \mathfrak{R} \) of all real numbers. Then we call \( \mathfrak{D} \) an absolute valued algebra if there is a function \( \phi(a) \) on \( \mathfrak{D} \) to \( \mathfrak{R} \) such that

\( \phi(0) = 0 \),
\( \phi(a) > 0 \) if \( a \neq 0 \),
\( \phi(ab) = \phi(a)\phi(b) \),
\( \phi(a+b) \leq \phi(a) + \phi(b) \),
\( \phi(\alpha a) = |\alpha|\phi(a) \) for every \( a \) and \( b \) of \( \mathfrak{D} \) and real \( \alpha \) where \( |\alpha| \) is the ordinary absolute value.

Then \( \phi(a\pm b) \geq |\phi(a) - \phi(b)| \). Also \( \mathfrak{D} \) must be a division algebra. For if \( a\neq 0 \), \( b\neq 0 \), then

\( \phi(a) > 0 \),
\( \phi(b) > 0 \),
\( \phi(ab) = \phi(a)\phi(b) > 0 \), \( ab\neq 0 \).

It is well known that the only alternative division algebras over \( \mathfrak{R} \) are the field \( \mathfrak{R} \), the field \( \mathfrak{C} \) of all complex numbers, the algebra \( \mathfrak{Q} \) of all real quaternions, and the eight dimensional algebra \( \mathfrak{B} \) of all real Cayley numbers. Each of these algebras has a conjugate operation \( x\to \bar{x} \) such that \( x\bar{x} \) is a positive real number, and it is easy to see that \( \phi(x) = \sqrt{x\bar{x}} \) defines an absolute value for each algebra. A known result then implies that \( \phi(a) \) is unique. We shall prove here that the alternative division algebras are the only absolute valued real algebras with a unity quantity. This characterizes the Cayley algebra as the only absolute valued real algebra with a unity quantity which is not associative. We shall also show that all absolute valued real algebras without a unity quantity are certain isotopes of \( \mathfrak{Q} \) and \( \mathfrak{B} \).

A real algebra \( \mathfrak{A} \) is called a normed algebra if the function \( \phi(a) \) is assumed to satisfy \[ \phi(ab) \leq \phi(a)\phi(b) \] rather than the equality. We shall show here that every real algebra is a normed algebra. Then there exist normed algebras which are not absolute valued.

A. A. Albert: “A structure theory for Jordan algebras,” Ann. Math. (2) 48 : 3 (July 1947), pp. 546–567. MR 0021546 Zbl 0029.01003 article

A. A. Albert: “On the power-associativity of rings,” Summa Brasil. Math. 2 : 2 (1948), pp. 21–32. MR 0026044 Zbl 0039.26403 article

A. A. Albert: “Absolute-valued algebraic algebras,” Bull. Am. Math. Soc. 55 : 8 (1949), pp. 763–768. MR 0030941 Zbl 0033.34901 article

An algebra \( \mathfrak{A} \) over a field \( \mathfrak{F} \) is a vector space over \( \mathfrak{F} \) which is closed with respect to a product \( xy \) which is linear in both \( x \) and \( y \). The product is not necessarily associative. Every element \( x \) of \( \mathfrak{A} \) generates a subalgebra \( \mathfrak{F}[x] \) of \( \mathfrak{A} \) and we call \( \mathfrak{A} \) an algebraic algebra if every \( \mathfrak{F}[x] \) is a finite-dimensional vector space over \( \mathfrak{F} \).

We have shown elsewhere [1947] that every absolute-valued real finite-dimensional algebra has dimension 1, 2, 4, or 8 and is either the field \( \mathfrak{R} \) of all real numbers, the complex field \( \mathfrak{C} \), the real quaternion algebra \( \mathfrak{Q} \), the real Cayley algebra \( \mathfrak{D} \), or certain isotopes without unity quantities of \( \mathfrak{Q} \) and \( \mathfrak{D} \). In the present paper we shall extend these results to algebraic algebras over \( \mathfrak{R} \) showing that every algebraic algebra over \( \mathfrak{R} \) with a unity quantity is finite-dimensional and so is one of the algebras listed above. The results are extended immediately to absolute-valued algebraic division algebras, that is, to algebras without unity quantities whose nonzero quantities form a quasigroup.

A. A. Albert: “A note on the exceptional Jordan algebra,” Proc. Nat. Acad. Sci. U. S. A. 36 : 7 (July 1950), pp. 372–374. MR 0035753 Zbl 0037.15904 article

An associative algebra \( \mathfrak{A} \) over a field \( \mathfrak{F} \) is a vector space over \( \mathfrak{F} \) together with an associative bilinear operation \( xy \). When the characteristic of \( \mathfrak{F} \) is not two we can use the same vector space and define a new algebra \( \mathfrak{A}^{(+)} \) relative to the operation \( \frac{1}{2}(xy + yx) \). This algebra is a Jordan algebra. Any Jordan algebra \( \mathfrak{J} \) is called a special Jordan algebra if \( \mathfrak{J} \) is isomorphic to a Jordan subalgebra of some \( \mathfrak{A}^{(+)} \).

In 1934 it was shown that the Jordan algebra \( \mathfrak{G} \) of all three-rowed Hermitian matrices with elements in the simple eight-dimensional Cayley algebra \( \mathfrak{C} \) is exceptional in the limited sense that \( \mathfrak{G} \) is not isomorphic to a subalgebra of a finite-dimensional \( \mathfrak{A}^{(+)} \). In the present note we shall give a simpler proof of the fact that \( \mathfrak{G} \) is not a special Jordan algebra and shall delete the restriction that \( \mathfrak{A} \) be finite dimensional. We shall assume that \( \mathfrak{G} \) is imbedded in an associative algebra \( \mathfrak{A} \) and shall then obtain a contradiction by showing that the enveloping associative algebra of \( \mathfrak{G} \) contains a subalgebra isomorphic to the non-associative algebra \( \mathfrak{T} \).

A. A. Albert: “A theory of power-associative commutative algebras,” Trans. Am. Math. Soc. 69 : 3 (November 1950), pp. 503–527. MR 0038959 Zbl 0039.26501 article

A. A. Albert: “On nonassociative division algebras,” Trans. Am. Math. Soc. 72 (1952), pp. 296–309. MR 0047027 Zbl 0046.03601 article

A. A. Albert: “On simple alternative rings,” Can. J. Math. 4 : 2 (1952), pp. 129–135. MR 0048420 Zbl 0046.25403 article

The only known simple alternative rings which are not associative are the Cayley algebras. Every such algebra has a scalar extension which is isomorphic over its center \( \mathsf{F} \) to the algebra \[ \mathsf{C} = e_{11}\mathsf{F} + e_{00}\mathsf{F} + \mathsf{C}_{10} + \mathsf{C}_{01} ,\] where \[ \mathsf{C}_{ij} = e_{ij}\mathsf{F} + f_{ij}\mathsf{F} + g_{ij}\mathsf{F} \qquad (i,j = 0,1;\ i \neq j) .\] The elements \( e_{11} \) and \( e_{00} \) are orthogonal idempotents and \[ e_{ii}x_{ij} = x_{ij}e_{jj} = x_{ij},\quad e_{jj}x_{ij} = x_{ij}e_{ii} = 0,\quad x_{ij}^2 = 0 \] for every \( x_{ij} \) of \( \mathsf{C}_{ij} \). The multiplication table of \( \mathsf{C} \) is then completed by the relations \begin{gather*} f_{10}g_{10} = e_{01},\quad g_{10}e_{10} = f_{01},\quad e_{10}f_{10} = g_{01},\\ g_{01}f_{01} = e_{10},\quad e_{01}g_{01} = f_{10},\quad f_{01}e_{01} = g_{10},\\ e_{ij}e_{ji} = f_{ij}f_{ji} = g_{ij}e_{ji} = e_{ii},\\ e_{ij}f_{ji} = e_{ij}g_{ji} = f_{ij}e_{ji} = f_{ij}g_{ji} = g_{ij}e_{ij} = g_{ij}f_{ij} = 0. \end{gather*}

Every simple alternative ring which contains an idempotent not its unity quantity is either associative or is the Cayley algebra \( \mathsf{C} \).

A. A. Albert and M. S. Frank: “Simple Lie algebras of characteristic \( p \),” Rend. Sem. Mat. Univ. Politec. Torino 14 (1954/1955), pp. 117–139. MR 0079222 Zbl 0065.26801 article

A. A. Albert and B. Muckenhoupt: “On matrices of trace zero,” Michigan Math. J. 4 : 1 (1957), pp. 1–3. MR 0083961 Zbl 0077.24304 article

A. A. Albert and N. Jacobson: “On reduced exceptional simple Jordan algebras,” Ann. Math. (2) 66 : 3 (November 1957), pp. 400–417. MR 0088487 Zbl 0079.04604 article

A. A. Albert: “A construction of exceptional Jordan division algebras,” Ann. Math. (2) 67 : 1 (January 1958), pp. 1–28. MR 0091946 Zbl 0079.04701 article

We shall construct a class of central simple exceptional Jordan algebras \( \mathfrak{H} \) over any field \( \mathfrak{F} \) of characteristic not two which we shall call cyclic Jordan algebras. Each such algebra \( \mathfrak{H} \) has an attached cyclic associative algebra \( \mathfrak{D} = (\mathfrak{K},S,\tau) \) of degree three over its center \( \mathfrak{F} \), and an attached Cayley algebra \( \mathfrak{C} \) over \( \mathfrak{K} \) having a nonsingular linear transformation \( T \) over \( \mathfrak{F} \) inducing \( S \) in \( \mathfrak{K} \) and such that \[ [(xy)T]\overline{g} = (xT)(yT\bar{g}), \qquad xT^3 = g^{-1}xg \] for \( g = gT \) in \( \mathfrak{C} \). We shall determine \( T \) completely in the case where \( \mathfrak{F}[g] \) is the direct sum of two copies of \( \mathfrak{F} \), and shall show that \( T \) is uniquely determined by a parameter \( \sigma \) in \( \mathfrak{F} \). When \( \mathfrak{D} \) is not a division algebra the cyclic algebra \( \mathfrak{H} \) is reduced. When \( \mathfrak{D} \) is a division algebra we can write \( \mathfrak{H} = \mathfrak{F}(\mathfrak{K},\tau,\sigma) \) and can show that every such \( \mathfrak{H} \) contains a subalgebra of dimension nine over \( \mathfrak{F} \) containing no singular elements. The norm form of \( \mathfrak{D} \) is a cubic form \( \Delta(x) \) in the nine coordinates of the general element \( x \) of \( \mathfrak{D} \), and we shall show that \( \mathfrak{H} \) contains a singular element if and only if \( \sigma = \Delta(x) \) for some \( x \) of \( \mathfrak{D} \). Also \( \mathfrak{H} \) is reduced if and only if \( \sigma = \Delta(x) \). It follows that every simple exceptional Jordan algebra over an algebraic number field is reduced. However, if \( \mathfrak{F}_0 \) is any field of characteristic not two such that there exists a cyclic associative division algebra \( \mathfrak{D}_0 = (\mathfrak{K}_0, S, \tau) \) over \( \mathfrak{F}_0 \), and \( \sigma \) is an indeterminate over \( \mathfrak{F}_0 \), the cyclic Jordan algebra \( \mathfrak{H} = \mathfrak{H}(\mathfrak{K}_{0\mathfrak{F}}, \tau, \sigma) \) over \( \mathfrak{F} = \mathfrak{F}_0(\sigma) \) is not reduced and contains no divisors of zero.

A. A. Albert and J. Thompson: “Two-element generation of the projective unimodular group,” Ill. J. Math. 3 : 3 (1959), pp. 421–439. MR 0106951 Zbl 0098.02302 article

Let \( \mathfrak{F} = \mathfrak{F}_q \) be the field of \( q = p^m \) elements, \( \mathfrak{M} = \mathfrak{M}(n,q) \) the multiplicative group of all \( n \)-rowed square matrices with elements in and determinant 1, and \( \mathfrak{N} = \mathfrak{N}(n,q) \) the subgroup of \( \mathfrak{M} \) consisting of its scalar matrices \( \rho I \) with \( \rho^n = 1 \). We assume, of course, that \( n > 1 \). Then \( \mathfrak{N} \) is a normal subgroup of \( \mathfrak{N} \), and the quotient group \[ \mathfrak{G}=\mathfrak{G}(n,q)=\mathfrak{M}/\mathfrak{N} \] is a well-known simple group called the projective unimodular group.

In [1930] H. R. Brahana gave a list of simple groups of orders less than 1,000,000. An examination of his list reveals the fact that every group there is generated by two elements, one of which has period (group order) two. The purpose of this paper is to prove the corresponding result for a general class of simple groups. We shall derive the following property.

The projective unimodular group is generated by two elements \( A\mathfrak{N} \) and \( B\mathfrak{N} \), where the coset \( A\mathfrak{N} \) has period two.

A. A. Albert and L. J. Paige: “On a homomorphism property of certain Jordan algebras,” Trans. Am. Math. Soc. 93 (1959), pp. 20–29. MR 0108524 Zbl 0089.02001 article

A. A. Albert: “On the collineation groups of certain non-desarguesian planes,” Port. Math. 18 (1959), pp. 207–224. MR 0130613 Zbl 0099.15203 article

A. A. Albert: “Finite division algebras and finite planes,” pp. 53–70 in Combinatorial analysis (Columbia University, New York, April 24–26, 1958). Edited by R. E. Bellman and M. Hall, Jr. Proceedings of Symposia in Applied Mathematics 10. American Mathematical Society (Providence, RI), 1960. MR 0116036 Zbl 0096.15003 incollection

Richard Ernest Bellman	Related
Marshall Hall, Jr.	Related

A. A. Albert: “Generalized twisted fields,” Pac. J. Math. 11 : 1 (1961), pp. 1–8. MR 0122850 Zbl 0154.27203 article

A. A. Albert: “On involutorial associative division algebras,” Scripta Math. 26 (1963), pp. 309–316. MR 0179202 Zbl 0147.28702 article

A. A. Albert: “On associative division algebras of prime degree,” Proc. Am. Math. Soc. 16 (1965), pp. 799–802. MR 0179203 Zbl 0135.07402 article

In 1938 Richard Brauer showed that, if \( \mathfrak{D} \) is an associative division algebra of degree five over its center \( \mathfrak{F} \), there is a field \( \mathfrak{K} \) of degree at most twelve over \( \mathfrak{F} \) such that the scalar extension \( \mathfrak{D}\times\mathfrak{K} \) is a cyclic algebra over \( \mathfrak{K} \). Indeed \[ \mathfrak{K}=\mathfrak{F}(\alpha_1,\alpha_2,\alpha_3) ,\] where \( \alpha_1 \) is a root of a quadratic equation over \( \mathfrak{F} \), \( \alpha_2 \) is a root of a quadratic equation over \( \mathfrak{F}(\alpha_1) \), and \( \alpha_3 \) is a root of a cubic equation over \( \mathfrak{F}(\alpha_1,\alpha_2) \). Since that time there has been no progress in the study of the structure of associative central division algebras of prime degree.

In view of Brauer’s result it seems reasonable, as a first step in the study of central division algebras \( \mathfrak{D} \) of prime degree \( p \) over \( \mathfrak{F} \), to consider the case where there is a quadratic field \( \mathfrak{K} \) over \( \mathfrak{F} \) such that \( \mathfrak{D}_{\mathfrak{K}} \) is cyclic. Then \[ \mathfrak{D}_{\mathfrak{K}}=\mathfrak{D}\times\mathfrak{K} \] has a subfield \( \mathfrak{Z} \) which is cyclic of degree \( p \) over \( \mathfrak{F} \). The simplest subcase is that where \( \mathfrak{Z} \) is actually normal, but, of course, not cyclic over \( \mathfrak{F} \). We shall treat this case under the assumption that \( \mathfrak{F} \) has characteristic \( p \), and shall prove that then \( \mathfrak{D} \) is a cyclic algebra over \( \mathfrak{F} \).

A. A. Albert: “On exceptional Jordan division algebras,” Pac. J. Math. 15 : 2 (1965), pp. 377–404. MR 0182647 Zbl 0135.07202 article

In 1957 the author gave a construction of a class, of central simple exceptional Jordan algebras \( \mathfrak{H} \), over any field \( \mathfrak{F} \) of characteristic not two, called cyclic Jordan algebras. The principal ingredients of this construction were the following:

A cyclic cubic field \( \mathfrak{K} \) with generating automorphism \( S \) over \( \mathfrak{F} \).
A Cayley algebra \( \mathfrak{C} \), with \( \mathfrak{K} \) as center, so that \( \mathfrak{C} \) has dimension eight over \( \mathfrak{K} \), and dimension 24 over \( \mathfrak{F} \).
A nonsingular linear transformation \( T \) over \( \mathfrak{F} \) of \( \mathfrak{C} \), which induces \( S \) in \( \mathfrak{K} \), and commutes with the conjugate operation of \( \mathfrak{C} \).
An element \( g \) in \( \mathfrak{C} \), and a nonzero element \( \gamma \) of \( \mathfrak{K} \), such that \[ g=gT \quad\text{and}\quad g\bar{g} = [\gamma(\gamma S)(\gamma S^2)]^{-1} .\] Thus \( g \) is nonsingular. Also the polynomial algebra \( \mathfrak{G}=\mathfrak{F}[g] \) is either a quadratic field over \( \mathfrak{F} \) or is the direct sum, \[ \mathfrak{G}=e_1\mathfrak{F}\oplus e_2\mathfrak{F} ,\] of two copies \( e_i\mathfrak{F} \) of \( \mathfrak{F} \).
The properties \[ [g(xy)T]=[g(xT)](yT) \quad\text{and}\quad xT^3 = g^{-1}xg ,\] for every \( x \) and \( y \) of \( \mathfrak{C} \).

In the present paper we shall give a general solution of the equations of (v), and shall determine \( T \) in terms of two parameters in \( \mathfrak{L}=\mathfrak{K}[g] \) satisfying some conditions of an arithmetic type. We shall also provide a special set of values of all of the parameters of our construction, and shall so provide a proof of the existence of cyclic Jordan division algebras with attached Cayley algebra \( \mathfrak{C} \) a division algebra.

The existence of a transformation \( T \) with the two properties of (v) for some element \( g = gT \), in the Cayley algebra \( \mathfrak{C} \) which satisfies (iv), was demonstrated by the author in the paper [Albert 1958] only in the case where \( \mathfrak{G} \) is not a field, and consequently \( \mathfrak{C} \) is a split algebra. In that case it was proved that cyclic Jordan division algebras do exist, for certain kinds of fields \( \mathfrak{F} \). Thus the case where \( \mathfrak{G} \) is a field, and \( \mathfrak{C} \) may possibly be a division algebra, remained.

A. A. Albert: “New results on associative division algebras,” J. Algebra 5 : 1 (January 1967), pp. 110–132. MR 0202757 Zbl 0144.02503 article

It is well known that every central division algebra of degree two or three is a cyclic algebra, and that every algebra of degree four is a crossed product. It is also known that there exist noncyclic algebras of degree four. Very little is known about algebras of degree \( n > 4 \).

The principal question in the theory of associative division algebras is then the question of the existence of algebras which are not crossed products. It is thus natural to discuss algebras of prime degree \( p \), a case where every crossed product is cyclic.

We shall address ourselves here to a study of the nature of division algebras \( \mathfrak{D} \) of prime degree \( p \) over a field \( \mathfrak{F} \) of characteristic \( p \), and shall limit our discussion to the case where there exists a quadratic extension \( \mathfrak{K} \) of \( \mathfrak{F} \) such that the extension \( \mathfrak{D}_0 = \mathfrak{D}\times\mathfrak{K} \) is a cyclic algebra.

A. A. Albert: “On associative division algebras,” Bull. Am. Math. Soc. 74 (1968), pp. 438–454. Retiring Presidential Address delivered at the 74th annual meeting of the AMS on January 23, 1967, in San Francisco. MR 0222114 Zbl 0157.08001 article

We shall study the structure of a central division algebra \( \mathfrak{D} \), of odd prime degree \( p \) over any field \( \mathfrak{F} \) of characteristic \( p \), which has the property that there exists a quadratic extension field \( \mathfrak{K} \) of \( \mathfrak{F} \) such that the algebra \( \mathfrak{D}_0 = \mathfrak{D}\times\mathfrak{K} \) is cyclic over \( \mathfrak{K} \). We shall obtain a simplified version of the condition that a cyclic algebra \( \mathfrak{D}_0 \), of degree \( p \) over \( \mathfrak{K} \), shall possess the factorization property \( \mathfrak{D}_0 = \mathfrak{D}\times\mathfrak{K} \). We shall also derive a new sufficient condition that such a \( \mathfrak{D} \) shall be cyclic over \( \mathfrak{F} \), and shall present a large class of our algebras \( \mathfrak{D}_0 \) which satisfy this condition.

A. A. Albert: “A note on certain cyclic algebras,” J. Algebra 14 : 1 (September 1970), pp. 70–72. MR 0251066 Zbl 0186.06901 article

A. A. Albert: “Tensor products of quaternion algebras,” Proc. Am. Math. Soc. 35 : 1 (September 1972), pp. 65–66. MR 0297803 Zbl 0263.16012 article

year	title	people

			clear

A. Adrian Albert

Abraham Adrian Albert, 1905–1972

Filter the Bibliography List