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Week of...
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Links
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1
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Jan 10
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About, Notes, HW1
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2
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Jan 17
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HW2, Notes
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3
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Jan 24
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HW3, Photo, Notes
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4
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Jan 31
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HW4, Notes
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5
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Feb 7
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HW5, Notes
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6
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Feb 14
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On TT, Notes
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R
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Feb 21
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Reading week
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7
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Feb 28
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Term Test
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8
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Mar 7
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HW6, Notes
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9
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Mar 14
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HW7, Notes
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10
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Mar 21
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HW8, E8, Notes
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11
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Mar 28
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HW9, Notes
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12
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Apr 4
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HW10, Notes
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13
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Apr 11
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Notes, PM
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S
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Apr 16-20
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Study Period
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F
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Apr 24
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Final
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![07-401 Class Photo.jpg](/images/thumb/d/d9/07-401_Class_Photo.jpg/180px-07-401_Class_Photo.jpg) Add your name / see who's in!
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Register of Good Deeds
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In Preparation
The information below is preliminary and cannot be trusted! (v)
The Fundamental Theorem of Galois Theory
It seems we will not have time to prove the Fundamental Theorem of Galois Theory in full. Thus this note is about what we will be missing. The statement appearing here, which is a weak version of the full theorem, is taken from Gallian's book and is meant to match our discussion in class. The proof is taken from Hungerford's book, except modified to fit our notations and conventions and simplified as per our weakened requirements.
Here and everywhere below our base field
will be a field of characteristic 0.
Statement
Theorem. Let
be a splitting field over
. Then there is a correspondence between the set
of intermediate field extensions
lying between
and
and the set
of subgroups
of the Galois group
of the original extension
:
.
The bijection is given by mapping every intermediate extension
to the subgroup
of elements in
that preserve
,
,
and reversely, by mapping every subgroup
of
to its fixed field
:
.
Furthermore, this correspondence has the following further properties:
- It is inclusion-reversing: if
then
and if
then
.
- It is degree/index respecting:
and
.
- Splitting fields correspond to normal subgroups: If
in
is a splitting field then
is normal in
and
.
Lemmas
The two lemmas below belong to earlier chapters but we skipped them in class.
The Primitive Element Theorem
The celebrated "Primitive Element Theorem" is just a lemma for us:
Lemma. Let
and
be algebraic elements of some extension
of
. Then there exists a single element
of
so that
. (And so by induction, every finite extension of
is "simple", meaning, is generated by a single element, called "a primitive element" for that extension).
Proof. See the proof of Theorem 21.6 on page 375 of Gallian's book.
Splitting Fields are Good at Splitting