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

176

THEORY OF EQUATIONS

162. Theorem of Lagrange as generalized by Galois. Any
n.tmber in the Galois domain which admits the substitutions of
the group Q is contained in the domain (,f).
In ~ 161 we saw that ilT, a function which belongs to Q,
assumed the following distinct values, when operated on by the
substitutions of P:,,
A, J,...,:;_i. T
Let MP' be any function in Q of the roots ca, *.., _.. which
admits the substitutions of Q. Let any substitution of P
which changes Minto M1i, change M' into M'i, then we get the
following values, corresponding to those in I,.M',.l, ", M's_l. II
These are not necessarily distinct.
Accordingly when upon the series of numbers I and II we
operate with a substitution of P, there occurs a permutation in
each series, but such that if Mi changes to M,., then M'i changes
to M'..
Defining +)(y) as in ~ 161, consider the function
~(y) - (y)f(y( X.. ' +which is an integral function of y of the (j - l)th degree.
This function is invariant under all substitutions of P. Hence
it is a function in 2. Take y = M. Remembering that +)(y)
has no equal roots, we have (reasoning as in ~ 142)
where q)' indicates the first differential coefficient of 4( with
respect to y. Thus M' is a number in the domain 0(M).
Ex. 1. Find the value of a root a of the equation x2 + 2 = 0 in terms
of a - al, it being given that P = 1, (aal).
If we take Q = 1, we see that M=_ a - al is a function which belongs
to Q and that W]/t _ a is a function which admits Q. We find M1 _ al -- a,