An introduction to the modern theory of equations, by Florian Cajori.

THE ALGEBRAIC SOLUTION OF EQUATIONS 229

usually very laborious even in the case of the quintic. Inasmuch as Bring, in 1786, and Jerrard, in 1834, were able to transform the general quintic to the form x1 + cx + d = 0 (for
this transformation, see Netto's Algebra, Vol. I, pp. 124, 125),
it is of interest to compute F(y) = 0 for this special form.
Ex. 1. Find the condition that the equation x5 + cx + d - 0, when
irreducible, shall be metacyclic.
Referring to ~ 104, we see that for the quintic the metacyclic group of
the highest order G is (abcde)2o. As a function belonging to this group
select (following C. Runge, Acta Math. 7 (1885), p. 173) y2, where
y coal + ala2c + a23 — + t34 +- C4g0o - 0a02 - a2a4 - a4i1 - a1a3 - 3a0O.
Here j - 6 and F(I/) = 0 is a resolvent equation of the sixth degree.
We find it convenient to consider y itself, which is not a inetacyclic function. Operated upon by the synmmetric group, y yields twelve values, of
which six differ rom the other rsix simply in sign. Let one set of six
values be Yl, Y2, *., y0. Also let the equation of which they are roots be
y6 + Caly5 + a2Y + a3y3 + a4y2 + a5y + a6 =0. I
Its coefficients a1, a2, *.., aC are not necessarily rational numbers, but
they are symmetric functions of yl, '", ye. Consider y/,.., y, as functions of ao,..., _n, and operate upon them with the alternating group;
the values yl,..., ye are merely permuted among themselves. Substitutions which do not belong to the alternating group bring abou aa change
in sign. The coefficients al, a2, *.., a6 are therefore either symmetric or
alternating functions of ao,..., a,_. Of these a2, a4, aG are symmetric
functions because, being homogeneous functions of even degree, they are
not affected by changes of signs in Yl, ye, y*-, YG. On the other hand,
ar, a3, a5 are alternating functions of ao, a1,..,,-1, being homogeneous
functions of odd degree.
If D is the discriminant of the quintic, then V\I is a function of
ao,..., n-1 belonging to the alternating group. Ience the coefficients
al, as, as are of the form mn\/D, m2z/D, ms31/D, where mil, mi2, m3 are
symmetric integral functions. With respect to ao, al,...,, n-l, it is seen
that al is of the second degree. But a, is also of the form mzl/D, where
in1 is integral and x/D is of the tenth degree. Hence we must have
mi = 0. Similarly, as being of the sixth degree, yields is - 0. On the
other hand, a5 and -/D are both of the tenth degree. Write a5 = m/D.
Equation I becomes

V6 + aC2Y' + a4Y2 + in VD-y + a6 = 0.

II