320
7. FIELD EXTENSIONS – FIRST LOOK
These are generally known as
Cardano’s formulas
, but Cardano credits
Scipione del Ferro and N. Tartaglia and for their discovery (prior to 1535).
What we will be concerned with here is the structure of the
field exten
sion
K
K.˛
1
; ˛
2
; ˛
3
/
and the symmetry of the roots.
Here is some general terminology and notation: If
F
L
are fields,
we say that
F
is a
subfield
of
L
or that
L
is a
field extension
of
F
. If
F
L
is a field extension and
S
L
is any subset, then
F.S/
denotes the
smallest subfield of
L
that contains
F
and
S
. If
F
L
is a field extension
and
g.x/
2
F OExŁ
has a complete set of roots in
L
(i.e.,
g.x/
factors into
linear factors in
LOExŁ
), then the smallest subfield of
L
containing
F
and
the roots of
g.x/
in
L
is called a
splitting field
of
g.x/
over
F
.
Returning to our more particular situation,
K.˛
1
; ˛
2
; ˛
3
/
is the split
ting field (in
C
) of the cubic polynomial
f .x/
2
KOExŁ
.
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 Fall '08
 EVERAGE
 Algebra, Polynomials, Formulas, Elementary algebra, Complex number, Cardano

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