Colloquium publications.

INVARIANTS AND NUMBER THEORY.

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tion (7) if
(19) P(A0o, A, *., Ar) A-P(ao, a,,..., a.) (mod p),
identically as to ao, * * *, ar, after the A's have been replaced by
their values (18) in terms of the ai. If P is invariant modulo p
under all transformations (7), it is called a formal invariant
modulo p of f.
The term formal is here used in connection with a form f whose
coefficients are arbitrary variables in contrast to the case, treated
in the earlier Lectures, in which the coefficients are undetermined integers taken modulo p. In the latter case, (19) necessarily becomes an identical congruence in the a's only after the
exponent of each a is reduced to a value less than p by means
of Fermat's theorem aP a (mod p).
The functions (18) are linear in ao, * * *, ar. It is customary to
say that relations (18) define a linear transformation on ao, * * *, ar
which is induced by the binary transformation (7). Let r be
the group of all of the transformations (18) induced by the group
of all of the binary transformations (7). Making no further
use of the form f, we may state the above problem of the determination of the formal invariants of f in the following terms.
We desire a fundamental system of invariants of group r. This
problem is of the type proposed in ~ 1; the group r is a special
group of order a multiple of p. Here and below the term invariant is restricted to rational integral functions of a0, * * ar.
A theory of formal invariants has not been found. For no
form f has a fundamental system of formal invariants been
published. Some light is thrown upon this interesting but
difficult problem by the following complete treatment of a
binary quadratic form, first for the exceptional case p = 2 and
next for the case p> 2, and preliminary treatment of a binary
cubic form.
6. Formal Invariants Modulo 2 of a Binary Quadratic Form.Write
(20) f = ax2 + bxy + cy2,