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Splitting Fields
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Summary
. In this article we further develop field theory in Mizar [1], [2]: we prove existence and uniqueness of splitting fields. We define the splitting field of a polynomial
p
∈
F
[
X
] as the smallest field extension of
F
, in which
p
splits into linear factors. From this follows, that for a splitting field
E
of
p
we have
E
=
F
(
A
) where
A
is the set of
p
’s roots. Splitting fields are unique, however, only up to isomorphisms; to be more precise up to
F
-isomorphims i.e. isomorphisms
i
with
i|
F
= Id
F
. We prove that two splitting fields of
p
∈
F
[
X
] are
F
-isomorphic using the well-known technique [4], [3] of extending isomorphisms from
F
1
→
F
2
to
F
1
(
a
) →
F
2
(
b
) for
a
and
b
being algebraic over
F
1
and
F
2
, respectively.
Title: Splitting Fields
Description:
Summary
.
In this article we further develop field theory in Mizar [1], [2]: we prove existence and uniqueness of splitting fields.
We define the splitting field of a polynomial
p
∈
F
[
X
] as the smallest field extension of
F
, in which
p
splits into linear factors.
From this follows, that for a splitting field
E
of
p
we have
E
=
F
(
A
) where
A
is the set of
p
’s roots.
Splitting fields are unique, however, only up to isomorphisms; to be more precise up to
F
-isomorphims i.
e.
isomorphisms
i
with
i|
F
= Id
F
.
We prove that two splitting fields of
p
∈
F
[
X
] are
F
-isomorphic using the well-known technique [4], [3] of extending isomorphisms from
F
1
→
F
2
to
F
1
(
a
) →
F
2
(
b
) for
a
and
b
being algebraic over
F
1
and
F
2
, respectively.
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