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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
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
.