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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 goal of today's class is to prove (a weak but strong enough) form of the Fundamental Theorem of Galois Theory as follows:
Theorem. Let
be a field of characteristic 0 and 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
.