Index of videos for 330
Please remember that when we were together we had class-time on
section 13 and 14, including one day entirely devoted to a
sophisticated example of cosets.
Since we have begun remotely here are our videos:
14.9: An example of theorem
14.9 including extensive work with cosets.
14.11: An example of
theorem 14.11 similar to 14.9 but from a different perspective.
14.13: Proving the
equivalence of different ways to test for a subgroup to be normal.
15: Another example of
normal subgroups and factor groups.
15.18: Simple groups and a
proof of theorem 15.18 about maximal normal subgroups and simple groups.
15.20: Abelianisation by
quotients from the commutator subgroup are in this proof of theorem
15.20.
XM2 Review videos:
18.8: high-school level
proofs of sign and zero properties in rings.
18: An example of a curious
field.
19.1: Zero divisors and
other analysis of the ring Z_n.
19.2: Relationship between
field and integral domain.
19.15: A pretty easy result
about characteristic of a ring
20.1: Fermat's little theorem
is rather easy to prove with algebra.
20.6 and 20.8: Euler extends
Fermat's result by working with rings and not only fields.
20: Some examples of equations
in Z_n. (and a
little more because the video stopped a tiny bit too early)
21: Introduction to the Field
of Fractions, and the set of S, whose equivalence classes form elements of
F.
21: Finishing step 1 of
defining the elements in the Field of Fractions, and the important
step 2 of defining the operations for F.
21: Talking about, but mostly
not doing, step 3 of proving that the Field of Fractions is a field, and
step 4 of seeing what we mean by saying that D is inside of F.
21: Introduction to theorem
21.6 - a different proof from the text, proving that the Field of
Fractions is the "smallest" field containing D.
21: Most of theorem 21.6.
21: Corollary 21.9, a slightly
different proof.
22: Most of section 22.
Highlights: proving than polynomial multiplication is associative
and some introduction to the evaluation homomorphism.
23: Proof of the division
algorithm for polynomials, along with the root-factor theorem.
23: Proof of Corollary 23.6
about all finite subgroups of units of a field being cyclic, with a
surprised appearance (to me at least) of Corollary 23.5.
23: An introduction to
reducible and irreducible polynomials.
23: Some simple examples
before Gauß's lemma (proven here, not by Fraleigh!)
23: Eistenstein's Criteria
plus a bonus application to cyclotomic polynomials.
23: Unique factorisation and a
corrected prior error.