
Solving Equations of Degree Four or Less 215
Quadratic Equations
The general quadratic polynomial is
t
2
−s
1
t + s
2
Let the zeros be t
1
and t
2
. The Galois group S
2
consists of the identity and a map
interchanging t
1
and t
2
. By Hilbert’s Theorem 90, Theorem 18.18, there must ex-
ist an element which, when acted on by the nontrivial element of S
2
, is multiplied
by a primitive square root of 1; that is, by −1. Obviously t
1
−t2 has this property.
Therefore
(t
1
−t
2
)
2
is fixed by S
2
, so lies in K(s
1
,s
2
). By explicit calculation
(t
1
−t
2
)
2
= s
2
1
−4s
2
Hence
t
1
−t
2
= ±
q
s
2
1
−4s
2
t
1
+t
2
= s
1
and we have the familiar formula
t
1
,t
2
=
s
1
±
q
s
2
1
−4s
2
2
Cubic Equations
The general cubic polynomial ...