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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
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Mathematical Problems by David Hilbert
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,
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Mathematical Problems by David Hilbert
X1 = f1(x1, ... , xn),
X2 = f2(x1, ... , xn),
(S)
...
Xm = fm(x1, ... , xn).
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, ... [ Pobierz całość w formacie PDF ]