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

146

'THEIIO:IY OF EQUATIONS

2, it is evident that p,1,.., "p,, are also symmetric functions
in f of a(, (a, *.., a,; for, an interchange of, say t and a1, brings
about simply an interchange of NV and N1. Since the interchange of N- and NV, does not alter these functions, the interchange of ca and ac does not.
Now a1, a2, *.., a,, are the roots of the equation f(x)=O; hence
the coefficients P1, P1'".n Pl of D(y) = 0, being symmetric functions in 0f of a1, *.., (,,, may be expressed as functions in l of
the coefficients of f(x)= 0 (~ 70).
But by hypothesis the coefficients of f(x) =0 are numbers
belonging to the domain fX, hence the same thing is true of
pi., pn,. Thus q>(y) = 0 is an equation of the nth degree in Q,
having the roots X, VX1,.., An,_.
Ex. 1. As an illustration, let f (x) = xl + 1 = 0, then 0 = Q(i1) and the
roots are ~ ~ x/2(1 + i), + x/2(1 - i). If c = ~- /2(1 + i), the domain
2(1, a) consists of numbers a + ib, where a and b may be rational, or
irrational involving V/2. Let NV = c3 + a2 + c +1, then N = 1 + (1 + /2)i,
and the numbers conjugate to it are,
N = 1 + (1 + /2)i, V2 = 1 (1 + V2)i,
Ni = 1 + (1 - /2)i, 3 = 1 (1 -2)i,
and (y) =(y-N)(y- -N)(y - X-2)(y - X3)
=4 _4y3+ 12 y2 - 16 y + 8 0.
Thus, AT and the numbers conjugate to it are roots of an algebraic equation of the fourth degree in Q(1), that is, D(y) = 0 is an equation in the
same domain as f (x) = 0, and both are of the same degree.
Ex. 2. Show that 5, i, V2 are each numbers lying in the domain Q(a)
of Ex. 1, and that each is a root of some reducible equation of the
fourth degree.
137. Theorem. Every number of the domain 1f(,) can be
expressed as a founction in 02 of any primitive number N of the
domain Q(a).
Let N' be any number in Q2(,) and N', NtV, N',, *.,? N'_1 the
numbers conjugate to it. Let
4() - (X- N)(x - )... (X - Nn_),