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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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Class Plan
Some discussion of the term test and HW6.
Some discussion of our general plan.
Lecture notes
Extension Fields
Definition. An extension field
of
.
Theorem. For every non-constant polynomial
in
there is an extension
of
in which
has a zero.
Example
over
.
Example
over
.
Definition.
.
Theorem. If
is a root of an irreducible polynomial
, within some extension field
of
, then
, and
(here
) is a basis for
over
.
Corollary. In this case,
depends only on
.
Splitting Fields
Definition.
splits in
, a splitting field for
over
.
Theorem. A splitting field always exists.
Example.
over
.
Example. Factor
within its splitting field
.
Theorem. Any two splitting fields for
over
are isomorphic.
Lemma 1. If
irreducible over
,
an isomorphism,
a root of
(in some
),
a root of
in some
, then
.
Lemma 2. Isomorphisms can be extended to splitting fields.
Zeros of Irreducible Polynomials
(This section was not covered on March 7, parts of it will be covered later on).
Definition. The derivative of a polynomial.
Claim. The derivative operation is linear and satisfies Leibnitz's law.
Theorem.
has a multiple zero in some extension field of
iff
and
have a common factor of positive degree.
Lemma. The property of "being relatively prime" is preserved under extensions.
Theorem. Let
be irreducible. If
, then
has no multiple zeros in any extension of
. If
, then
has multiple zeros (in some extension) iff it is of the form
for some
.
Definition. A perfect field.
Theorem. A finite field is perfect.
Theorem. An irreducible polynomial over a perfect field has no multiple zeros (in any extension).
Theorem. Let
be irreducible and let
be the splitting field of
over
. Then in
all zeros of
have the same multiplicity.
Corollary.
as above must have the form
for some
and
.
Example.
is irreducible and has a single zero of multiplicity 2 within its splitting field over
.
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07-401 March 7 NOTES
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07-401 March 7 NOTES
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07-401 March 7 NOTES
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07-401 March 7 NOTES
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07-401 March 7 NOTES
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07-401 March 7 NOTES