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12.
Extension of Kroneker's theorem on abelian fields to any algebraic realm
of rationality
The
theorem that every abelian number field arises from the realm of rational
numbers by the composition of fields of roots of unity is due to
Kronecker. This fundamental theorem in the theory of integral equations
contains two statements, namely: First.
It answers the question as to the number and existence of those equations
which have a given degree, a given abelian group and a given discriminant
with respect to the realm of rational numbers. Second.
It states that the roots of such equations form a realm of algebraic
numbers which coincides with the realm obtained by assigning to the
argument z in the exponential function ei The
first statement is concerned with the question of the determination of
certain algebraic numbers by their groups and their branching. This
question corresponds, therefore, to the known problem of the
determination of algebraic functions corresponding to given Riemann
surfaces. The second statement furnishes the required numbers by
transcendental means, namely, by the exponential function ei Since
the realm of the imaginary quadratic number fields is the simplest after
the realm of rational numbers, the problem arises, to extend Kronecker's
theorem to this case. Kronecker himself has made the assertion that the
abelian equations in the realm of a quadratic field are given by the
equations of transformation of elliptic functions with singular moduli,
so that the elliptic function assumes here the same role as the
exponential function in the former case. The proof of Kronecker's
conjecture has not yet been furnished; but I believe that it must be
obtainable without very great difficulty on the basis of the theory of
complex multiplication developed by H. Weber25
with the help of the purely arithmetical theorems on class fields which I
have established. Finally,
the extension of Kronecker's theorem to the case that, in place of the
realm of rational numbers or of the imaginary quadratic field, any
algebraic field whatever is laid down as realm of rationality, seems
to me of the greatest importance. I regard this problem as one of the
most profound and far reaching in the theory of numbers and of functions.
The
problem is found to be accessible from many standpoints. I regard as the
most important key to the arithmetical part of this problem the general
law of reciprocity for residues of I-th powers within any given
number field. As
to the function-theoretical part of the problem, the investigator in this
attractive region will be guided by the remarkable analogies which are
noticeable between the theory of algebraic functions of one variable and
the theory of algebraic numbers. Hensel26 has proposed and investigated the analogue in the theory of algebraic
numbers to the development in power series of an algebraic function; and
Landsberg27
has treated the analogue of the Riemann-Roch theorem. The analogy between
the deficiency of a Riemann surface and that of the class number of a
field of numbers is also evident. Consider a Riemann surface of
deficiency p = 1 (to touch on the simplest case only) and on the
other hand a number field of class h = 2. To the proof of the
existence of an integral everywhere finite on the Riemann surface,
corresponds the proof of the existence of an integer a in the
number field such that the number
The
equation of Abel's theorem in the theory of algebraic functions
expresses, as is well known, the necessary and sufficient condition that
the points in question on the Riemann surface are the zero points of an
algebraic function belonging to the surface. The exact analogue of Abel's
theorem, in the theory of the number field of class h = 2, is the
equation of the law of quadratic reciprocity28
which
declares that the ideal j is then and only then a principal ideal
of the number field when the quadratic residue of the number a
with respect to the ideal j is positive. It
will be seen that in the problem just sketched the three fundamental
branches of mathematics, number theory, algebra and function theory, come
into closest touch with one another, and I am certain that the theory of
analytical functions of several variables in particular would be notably
enriched if one should succeed in finding and discussing those
functions which play the part for any algebraic number field
corresponding to that of the exponential function in the field of
rational numbers and of the elliptic modular functions in the imaginary
quadratic number field. Passing
to algebra, I shall mention a problem from the theory of equations and
one to which the theory of algebraic invariants has led me.
13.
Impossibility of the solution of the general equation of the 7-th degree
by means of functions of only two arguments
Nomography29
deals with the problem: to solve equations by means of drawings of
families of curves depending on an arbitrary parameter. It is seen at
once that every root of an equation whose coefficients depend upon only
two parameters, that is, every function of two independent variables, can
be represented in manifold ways according to the principle lying at the
foundation of nomography. Further, a large class of functions of three or
more variables can evidently be represented by this principle alone
without the use of variable elements, namely all those which can be
generated by forming first a function of two arguments, then equating
each of these arguments to a function of two arguments, next replacing
each of those arguments in their turn by a function of two arguments, and
so on, regarding as admissible any finite number of insertions of
functions of two arguments. So, for example, every rational function of
any number of arguments belongs to this class of functions constructed by
nomographic tables; for it can be generated by the processes of addition,
subtraction, multiplication and division and each of these processes
produces a function of only two arguments. One sees easily that the roots
of all equations which are solvable by radicals in the natural realm of
rationality belong to this class of functions; for here the extraction of
roots is adjoined to the four arithmetical operations and this, indeed,
presents a function of one argument only. Likewise the general equations
of the 5-th and 6-th degrees are solvable by suitable nomographic tables;
for, by means of Tschirnhausen transformations, which require only
extraction of roots, they can be reduced to a form where the coefficients
depend upon two parameters only. Now
it is probable that the root of the equation of the seventh degree is a
function of its coefficients which does not belong to this class of
functions capable of nomographic construction, i. e., that it
cannot be constructed by a finite number of insertions of functions of
two arguments. In order to prove this, the proof would be necessary that
the equation of the seventh degree f7 + xf3
+ yf2 + zf + 1 = 0 is not solvable with the
help of any continuous functions of only two arguments. I may be
allowed to add that I have satisfied myself by a rigorous process that
there exist analytical functions of three arguments x, y, z which
cannot be obtained by a finite chain of functions of only two arguments. By
employing auxiliary movable elements, nomography succeeds in constructing
functions of more than two arguments, as d'Ocagne has recently proved in
the case of the equation of the 7-th degree.30
14.
Proof of the finiteness of certain complete systems of functions
In
the theory of algebraic invariants, questions as to the finiteness of
complete systems of forms deserve, as it seems to me, particular
interest. L. Maurer31
has lately succeeded in extending the theorems on finiteness in invariant
theory proved by P. Gordan and myself, to the case where, instead of the
general projective group, any subgroup is chosen as the basis for the
definition of invariants. An
important step in this direction had been taken al ready by A. Hurwitz,32
who, by an ingenious process, succeeded in effecting the proof, in its
entire generality, of the finiteness of the system of orthogonal
invariants of an arbitrary ground form. The
study of the question as to the finiteness of invariants has led me to a
simple problem which includes that question as a particular case and
whose solution probably requires a decidedly more minutely detailed study
of the theory of elimination and of Kronecker's algebraic modular systems
than has yet been made. Let
a number m of integral rational functions Xl, X2,
... , Xm, of the n variables xl,
x2, ... , xn be given,
Every
rational integral combination of Xl, ... , Xm
must evidently always become, after substitution of the above
expressions, a rational integral function of xl, ... , xn.
Nevertheless, there may well be rational fractional functions of Xl,
... , Xm which, by the operation of the substitution S,
become integral functions in xl, ... , xn.
Every such rational function of Xl, ... , Xm,
which becomes integral in xl, ... , xn
after the application of the substitution S, I propose to call a relatively
integral function of Xl, ... , Xm.
Every integral function of Xl, ... , Xm
is evidently also relatively integral; further the sum, difference and
product of relative integral functions are themselves relatively
integral. The
resulting problem is now to decide whether it is always possible to
find a finite system of relatively integral function Xl,
... , Xm by which every other relatively integral
function of Xl, ... , Xm may
be expressed rationally and integrally. We
can formulate the problem still more simply if we introduce the idea of a
finite field of integrality. By a finite field of integrality I mean a
system of functions from which a finite number of functions can be
chosen, in terms of which all other functions of the system are
rationally and integrally expressible. Our problem amounts, then, to
this: to show that all relatively integral functions of any given domain
of rationality always constitute a finite field of integrality. It
naturally occurs to us also to refine the problem by restrictions drawn
from number theory, by assuming the coefficients of the given functions fl,
... , fm to be integers and including among the
relatively integral functions of Xl, ... , Xm
only such rational functions of these arguments as become, by the
application of the substitutions S, rational integral functions of
xl, ... , xn with rational integral
coefficients. The
following is a simple particular case of this refined problem: Let m
integral rational functions Xl, ... , Xm
of one variable x with integral rational coefficients, and a prime
number p be given. Consider the system of those integral rational
functions of x which can be expressed in the form G(Xl, ... , Xm) / ph, where
G is a rational integral function of the arguments Xl,
... , Xm and ph is any power of the
prime number p. Earlier investigations of mine33 show immediately that all such expressions for a fixed exponent h
form a finite domain of integrality. But the question here is whether the
same is true for all exponents h, i. e., whether a finite
number of such expressions can be chosen by means of which for every
exponent h every other expression of that form is integrally and
rationally expressible. From
the boundary region between algebra and geometry, I will mention two
problems. The one concerns enumerative geometry and the other the
topology of algebraic curves and surfaces. |
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