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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
FŒxŁ
has a complete set of roots in
L
(i.e.,
g.x/
factors into
linear factors in
LŒxŁ
), 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
KŒxŁ
.
One noticeable feature of this situation is that the roots of
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
 Fall '08
 EVERAGE
 Algebra, Formulas

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