THEORY OF NON-COMMUTATIVE POLYNOMIALS

Algebra und Zahlentheorie
Ferner erweist sich in Untersuchungen von Chevalley und mir die Theorie der z. A.
als das adäquate Werkzeug zum Aufbau der Klassenkörpertheorie im Kleinen.
Mittels Satz 3 konnte ich dann weiter den Uebergang zur Klassenkörpertheorie im
Großen für den Isomorphiesatz und das Artinsche Rez. Ges. in eleganter Weise vollziehen.
Schließlich erweist sich in Untersuchungen von Artin, E. Noether und mir Satz 3
(und überhaupt diese Methode) als kräftiges Hilfsmittel bei der Behandlung der
großen im Mittelpunkt der modernen Zahlentheorie stehenden Frage nach dem Zerlegungsgesetz in allg. galoisschen Zahlkörpern.
THEORY OF NON-COMMUTATIVE POLYNOMIALS
By OYSTEIN ORE, New Haven
The present paper contains some of the main properties of polynomials
(1)
f(x) = a0x» + axx* - 1 + , . . . . + an
with coefficients a . in a non-commutative field K. The sum of two polynomials (1)
is defined as the polynomial obtained by adding corresponding coefficients. For the
definition of the product there exists various possibilities. In non-commutative
algebras one usually considers the case, where the variable x is permutable with the
coefficients; a wider domain of applications is obtained however, if one only postulates that the degree of the product is always equal to the sum of the degrees of
the factors. The theory then also includes for instance the formal theory of linear
differential and difference equations and also of other functional equations.
Under this assumption one must have for a linear product
2)
xa = ax -f- a'
where a is an arbitrary element in K; from (2) the general product of polynomials (1)
can be defined. For the two fundamental operations, the conjugation a and the
differentiation a' one derives the rules
a --j- b = a -f- b, ab = a*b
(a -\ - b)' = a' + b', {ab)f = ab1 + a'b.
The first represents a homomorphismus in K, in the commutative case the second
obeys rules corresponding to ordinary differentiation.
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Algebra und Zahlentheorie
A right-hand Euclid algorithm always exists for two polynomials; a left-hand
algorithm exists only when the conjugation satisfies a certain condition. From the
right-hand algorithm the existence of a right-hand greatest common divisor and
least common multiplum follows.
The notion of the transform gf (x) g—1 of a polynomial f (x) by another g (x) is
then introduced; it contains f. inst. the notion of similarity introduced for linear
differential expressions by Poincaré. Among the various properties of the transform I shall only mention the following: When a product a (x) b (x) is divisible
by c (x), and b (x) relatively prime to c (x) then a (x) is divisible by be (x) &—1.
The notion of transforms is particularly useful in the formulation and proof of
the various theorems on the representation of non-commutative polynomials, which
form the principal object of this paper. Four main representations are considered:
The first is the representation as a product of prime factors; the second is the
representation as a product of reducible factors, both corresponding to theorems
of Loewy in the special case of linear differential equations. The third representation
includes as a special case the representation studied by Krull for differential polynomials, while the proofs are considerably simpler. The fourth representation is
new. I finally observe that this general theory has been carried further than the
special theory of representation for differential- and difference equations, so that
various new results can be derived also for these cases.
IDEAL- UND BEWERTUNGSBEGRIFF IN DER
ARITHMETIK DER KOMMUTATIVEN INTEGRITÄTSBEREICHE
Von W. KRULL, Erlangen
Im Ring der ganzen rationalen Zahlen liefert die Primfaktorzerlegung das Teilbarkeitsgesetz für die Elemente. In den Hauptordnungen der endlichen algebraischen
Zahlkörper, in denen für die Elemente i*a keine eindeutige Zerlegung in Primfaktoren mehr besteht, hat das Teilbarkeitsproblem vor allem zwei klassische Lösungen gefunden. Dedekind führt neben den Elementen die Ideale ein, und erreicht
durch geeignete Definition des Ideals und der Idealmultiplikation, daß wenigstens
jedes Ideal eindeutig in endlich viele Primidealfaktoren zerlegt werden kann. Hensel
trennt die Primstellen durch Einführung der ^-adischen und ;r-adiscfien Zahlen und
vermeidet so den Idealbegriff. Arbeitet man mit dem Begriff der Bewertung, und
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