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Abstract Algebra II (Fall 2024) |
Lecture Notes |
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| Notes | Video | |
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8/19/2024: We discussed facts about groups in general and permutation groups. Notes from 8/19/24 |
Video from 8/19/24 | |
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8/21/2024: We discussed permutation groups, homomorphisms and isomorphisms, normal subgroups, and the first isomorphism theorem. Notes from 8/21/24 |
Video from 8/21/24 | |
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8/23/2024: We discussed cosets, Lagrange's Theorem, and group actions on sets. Notes from 8/23/24 |
Video from 8/23/24 | |
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8/26/2024: We discussed examples of group actions on sets. Notes from 8/26/24 |
Video from 8/26/24 | |
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8/28/2024: We discussed orbits, stabilizers, conjugacy classes of permutations, centralizers, and the orbit stabilizer theorem. Notes from 8/28/24 |
Video from 8/28/24 | |
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8/30/2024: We proved the orbit stabilizer theorem and sketched the proof of Cayley's theorem. We defined the center, centralizers of subsets, and normalizers of subsets of a group. See notes for a correction. Corrected Notes from 8/30/24 |
Video from 8/30/24 | |
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9/4/2024: We discussed cojugacy, normalizers, and the class equation. We also stated the Fundamental Theorem of Finitely Generated Abelian Groups. Notes from 9/4/24 |
Video from 9/4/24 | |
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9/6/2024: We prove the result about the relation between conjugacy classes and centralizers, and we proved the First Sylow Theorem. Notes from 9/6/24 |
Video from 9/6/24 | |
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9/9/2024: We proved a fact about counting elements of products of subgroups, and we stated the second and third Sylow theorems. We use that and the class equation to examine possibilities of isomorphism classes of groups of various orders. Notes from 9/9/24 |
Video from 9/9/24 | |
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9/11/2024: We classified groups of order 6 and calculated conditions that a group element is in the centralizer of a certain subgroup of $S_7$. Notes from 9/11/24 |
Video from 9/11/24 | |
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9/13/2024: We did examples of calculating normalizers, centralizers, and isomorphism classes of finite groups. Notes from 9/13/24 |
Video from 9/13/24 | |
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9/18/2024: We discussed commutative and noncommutative rings, with and without unity. We calculated the group of units for an example ring, and we discussed integral domains. Notes from 9/18/24 |
Video from 9/18/24 | |
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9/20/2024: We discussed ring homomorphisms, ideals, ideals generated by a finite set of elements, fields of quotients of integral domains, prime ideals, maximal ideals, and principal ideals. Notes from 9/20/24 |
Video from 9/20/24 | |
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9/23/2024: We discussed an example of a nonprincipal ideal that is maximal and prime, and we proved results connecting properties of ideals with properties of the corresponding factor rings. Notes from 9/23/24 |
Video from 9/23/24 | |
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9/25/2024: We proved results about rings and ideals of various types and the associated factor rings. We found isomorphism types of a few examples of factor rings. Notes from 9/25/24 |
Video from 9/25/24 | |
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9/27/2024: We discussed irreducibles, associates, and unique factorization domains. Notes from 9/27/24 |
Video from 9/27/24 | |
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9/30/2024: We discussed unique factorization domains, principal ideal domains, and Euclidean domains. Notes from 9/30/24 (corrected 10-6) |
Video from 9/30/24 | |
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10/2/2024: We discussed methods of factoring polynomials and irreducibility tests. We proved prime elements in an integral domain are always irreducible, and irreducible elements in principal ideal domains are prime. We also defined the ascending chain condition for commutative rings. Notes from 10/2/24 |
Video from 10/2/24 | |
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10/3/2024: We discussed the ascending chain condition (ACC) further and noted that every PID satisfies the ACC. Using this, we are able to show that every PID is a UFD. We also saw that every Euclidean domain is a PID. We displayed examples of ID's, UFD's, PID's, ED's, and Fields. Notes from 10/3/24 |
Video from 10/3/24 | |
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10/7/2024: We discussed the content of polynomials with integer coefficients,
and the fact that the product of primitive polynomials is primitive. We also mentioned some other facts about polynomial rings, such as the fact that if $R$ is a UFD, then $R[x]$ is a UFD. Notes from 10/7/24 |
Video from 10/7/24 | |
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10/14/2024: We began discussing extension fields and roots of polynomials. Notes from 10/14/24 |
Video from 10/14/24 | |
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10/16/2024: We found minimal polynomials of elements of extension fields. We proved that the minimal degree monic polynomial for which an element is a root is unique and is irreducible. Notes from 10/16/24 |
Video from 10/16/24 | |
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10/18/2024: We finished an example and discussed theorems about simple extensions and irreducible polynomials. Notes from 10/18/24 |
Video from 10/18/24 | |
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10/21/2024: We discussed simple algebraic extensions as vector spaces over the
base field. Notes from 10/21/24 |
Video from 10/21/24 | |
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10/23/2024: We discussed splitting fields of polynomials. Notes from 10/23/24 |
Video from 10/23/24 | |
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10/25/2024: We discussed characteristics of rings and characterizations of polynomials with multiple roots
over extension fields. Notes from 10/25/24 |
Video from 10/25/24 | |
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10/28/2024: We discussed finite field extensions, algebraic extensions, and the Tower Theorem. Notes from 10/28/24 |
Video from 10/28/24 | |
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10/30/2024: We talked about algebraic closures of fields in extension fields and about algebraically closed fields.
Using complex analysis, we proved the Fundamental Theorem of Algebra, that $\mathbb{C}$ is algebraically closed. Notes from 10/30/24 |
Video from 10/30/24 | |
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11/1/2024: We discussed constructible numbers and proved certain planar constructions with unmarked straight edge and
compass are impossible. We also showed that finite fields contain $\mathbb{Z}_p$ as a subfield for some prime $p$,
and they have order $p^n$ for some positive integer $n$. Edited Notes from 11/1/24 |
Video from 11/1/24 | |
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11/4/2024: We proved more facts about finite fields and their extension fields. Corrected Notes from 11/4/24 |
Video from 11/4/24 | |
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11/6/2024: We proved all fields of a given finite order are isomorphic,
and we discussed conjugate elements of extension fields and specific
isomorphisms of extension fields. Notes from 11/6/24 |
Video from 11/6/24 | |
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11/8/2024: We proved facts about automorphisms and fixed fields. Notes from 11/8/24 |
Video from 11/8/24 | |
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11/13/2024: We examined the isomorphism extension theorem, the Frobenius automorphism, and
normal extensions. Notes from 11/13/24 |
Video from 11/13/24 | |
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11/15/2024: We learned more about normal and separable extensions of fields. Notes from 11/15/24 |
Video from 11/15/24 | |
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11/18/2024: We learned about perfect fields and the Fundamental Theorem of Galois Theory. Notes from 11/18/24 |
Video from 11/18/24 | |
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11/20/2024: We worked on finding the Galois group of extensions by roots of unity. Notes from 11/20/24 |
Video from 11/20/24 | |
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11/22/2024: In an example, we saw the Fundamental Theorem of Galois Theory in action. Notes from 11/22/24 |
Video from 11/22/24 | |
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12/2/2024: We calculated the Galois groups and corresponding subfields of a particular field
extension of $\mathbb{Q}$. Notes from 12/2/24 |
Video from 12/2/24 | |
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12/4/2024: We discussed solvable groups and the proof that some fifth degree polynomials are not solvable by radicals. Notes from 12/4/24 |
Video from 12/4/24 | |