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

CHAPTER XV

REDUCTION OF THE GALOIS RESOLVENT BY ADJUNCTION
160. Definition of M. Let the Galois group P (of the
order p) of the equationf(x) = 0, having the roots a, aj, *.., al..1,
possess a sub-group Q of the order q, where p = qj, j being the
index of Q under P. For the purposes of the theorems in
succeeding chapters, we define M nearly as in ~ 159.
Let 1M be any fuiction ilZ Q of the roots a, *.., a,_n- which belongs
to Q as a sub-group of P (~ 111).
161. Theorem. By operating 2po) of M with the substitutions of
P we obtain j distinct values of MW which are roots of an irreducible
eqtuation of the jth degree in 2.
If t is a substitution of the Galois group P which does not
occur in the sub-group Q, and if s, s,, *.., sq_ be the substitutions
of Q, then by the definition of a group,
st, sit, *... Sq_-lt I
are all substitutions of P. But the substitutions srt in I, when
applied to M, all produce the same effect, for in any case we may
operate with the product st by first operating with s,. and then
upon the result with t. By hypothesis, operating with s, upon
M produces no change whatever, hence s,t produces always only
the result due to t alone.
By hypothesis it follows that, as t does not occur in the subgroup Q, t operated upon M gives us a new value Aft.
From ~ 106 we see that there are as many sets of substitutions I in the group P as q is contained in p; namely, j sets.
The substitutions of any one set applied to M all give the same
value for M, but no two sets yield the same value.
174