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

172

THEOlY OF EQUATIONS

assumptions are justified by the fact that the left member of each equation is a function which belongs to G8(4), ~ 154. We get
(a - C2)2 = 4(c + vb), (1 - 3))2 = 4(c - /b),
a - al + a2 - 3 = 4 d-Vb,
a( + a + a2 + s3 — 4 bl.
Hence a = bl + dVb + /c + V/b, ta = bl + cZIb - Vc + b,
a1=)b1-cdv/7 +Vc -Vb,3= -b1-d/b -Vc-vb.
Diminishing each root by bl and forming the quartic, we obtain the
result
result y4 _ 2(b2 )y2 _ 4 bdy + (bd2 - c)2 - b = 0.
Ex. 10. The quartic whose Galois group is G4(4)III is the reducible
equation,
X4 - 2(c2 + d)x2 - 4 cex + (c2 -d + e) (c2 - d - e)= 0,
where (d + e) and (d - e) are not perfect squares.
Derive this by assuming
Ct1 +- aL2 - (3 - (C4: 4 c,
(a1 - ca2)2 + (a3 - t4)2 = 8 d,
cal - as 2 - ga)o. = s e.
Ex. 11. Find a general expression for equations of the fourth degree
having the Galois group G4(4)I. Use the functions
(al - ia2 -( a3 + ia4)4,
(a1 + ia2 - a3 - ita4)4,
(a1 - C2 + X3 - a4)2,
(a1 - ia2 - a3 + iC4) (al + ia2 - a3 - i4),
(a1 - a2 + as - a4) ((1 - ia2 - a3 + ia4)2,
and impose upon the letters which appear in the expressions for the coefficients of the quartic no other conditions than that they shall be rational
and one of them shall not be a perfect fourth power. See Ex. 3, ~ 176.
Ex. 12. Show that, if the roots of the cubic in Ex. 11, ~ 71, are all
rational, the Galois group of the quartic having the roots a, /, y, 3 is
either G4(4)II or one of its sub-groups.
Consider
(a" + Y8)- _(a7 + 13).