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318
7. FIELD EXTENSIONS – FIRST LOOK
radicals and, in doing so, radically changed the nature of algebra. Galois
associated with a polynomial
p.x/
over a ﬁeld
K
a canonical smallest
ﬁeld
L
containing
K
in which the polynomial has a complete set of roots
and, moreover, a canonical group of symmetries of
L
, which acts on the
roots of
p.x/
.
Galois’s brilliant idea was to study the polynomial equation
p.x/
D
0
by means of this symmetry group. In particular, he showed that
solvability of the equation by radicals corresponded to a certain property
of the group, which also became known as
solvability
. The
Galois group
associated to a polynomial equation of degree
n
is always a subgroup of
the permutation group
S
n
. It turns out that subgroups of
S
n
for
n
±
4
are
solvable, but
S
5
is not solvable. In Galois’s theory, the nonsolvability of
S
5
implies the impossibility of an analogue of the quadratic formula for
equations of degree 5.
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 Fall '08
 EVERAGE
 Algebra, Radicals

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