17. EXPRESSION OF DEFINITE FORMS BY SQUARES.
A rational integral function or form in any number of variables with real coefficients such that it becomes negative for no real values of these variables, is said to bedefinite. The system of all definite forms is invariant with respect to the operations of addition and multiplication, but the quotient of two definite forms—in caseit should be an integral function of the variables—is also a definite form. The square of any form is evidently always a definite form. But since, as I have shown,[38]not every definite form can be compounded by addition from squares of forms, the question arises—which I have answered affirmatively for ternary forms[39]—whether every definite form may not be expressed as a quotient of sums of squares of forms. At the same time it is desirable, for certain questions as to the possibility of certain geometrical constructions, to know whether the coefficients of the forms to be used in the expression may always be taken from the realm of rationality given by the coefficients of the form represented.[40]
I mention one more geometrical problem:
[38]Math. Annalen, vol. 32.
[38]Math. Annalen, vol. 32.
[39]Acta Mathematica, vol. 17.
[39]Acta Mathematica, vol. 17.
[40]Cf. Hilbert: Grunglagen der Geometrie, Leipzig, 1899, Chap. 7 and in particular § 38.
[40]Cf. Hilbert: Grunglagen der Geometrie, Leipzig, 1899, Chap. 7 and in particular § 38.