Abstract Algebra: Groups, Rings & Fields

Video Lectures

Displaying all 83 video lectures.
Lecture 1
Mystery Division Problem
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Mystery Division Problem
Abstract Algebra: A pattern is given for a long division problem. That is, only the lengths of digits used are given. We determine the divisor, dividend, and quotient.
I. Group Theory
Lecture 2
GT1. Definition of Group
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GT1. Definition of Group
Abstract Algebra: We introduce the notion of a group and describe basic properties. Examples given include familiar abelian groups and the symmetric groups.

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Lecture 3
GT1.1. Example of Group Inverse
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GT1.1. Example of Group Inverse
Abstract Algebra: We note some useful rules for working with inverses in groups. 1) It is enough to find an inverse on one side only, 2) inverses are unique, and 3) the inverse of a product is the reversed product of the inverses.

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Lecture 4
Order 2 Elements in Finite Group
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Order 2 Elements in Finite Group
Abstract Algebra: Let G be a finite group. (1) If |G| is even, show that G has an odd number of elements of order 2. (2) If G is abelian, we compute the sum of the elements of the group (where group multiplication is written as addition).
Lecture 5
Example of Group Cancellation Law
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Example of Group Cancellation Law
Abstract Algebra: Let G be a nonempty finite set with multiplication. Suppose the multiplication is associative and satisfies cancellation laws on both sides. Show that G is a group.
Lecture 6
GT2. Definition of Subgroup
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GT2. Definition of Subgroup
Abstract Algebra: We define the notion of a subgroup and provide various examples. We also consider cyclic subgroups and subgroups generated by subsets in a given group G. Example include A4 and D8.

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Lecture 7
Example of Group: GL(2, R) (1 of 3)
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Example of Group: GL(2, R) (1 of 3)
Abstract Algebra: Let G = GL(2,R) be the set of real 2x2 invertible matrices. In this first part, we show that G is a group. Using the identity det(AB)=det(A)det(B), we give an indication of how to extend to nxn invertible matrices.
Lecture 8
GT3. Cosets and Lagrange's Theorem
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GT3. Cosets and Lagrange's Theorem
Abstract Algebra: Let G be a group with subgroup H. We define an equivalence relation on G that partitions G into left cosets. We use this partition to prove Lagrange's Theorem and its corollary.

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Lecture 9
GT4. Normal Subgroups and Quotient Groups
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GT4. Normal Subgroups and Quotient Groups
Abstract Algebra: We define normal subgroups and show that, in this case, the space of cosets carries a group structure, the quotient group. Example include S3, the modular integers, and Q/Z.

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Lecture 10
GT5. Index 2 Theorem and Dihedral Groups
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GT5. Index 2 Theorem and Dihedral Groups
Abstract Algebra: We state and prove the Index Two Theorem for finding normal subgroup and list several examples. These include S3, A4, and the symmetry groups for the regular n-gon, D_2n. We give several presentations of the latter groups and calculate the center.

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Lecture 11
Example of Group: GL(2,R) (2 of 3)
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Example of Group: GL(2,R) (2 of 3)
Abstract Algebra: Let G = GL(2, R) be the group of real 2 x2 invertible matrices, and let H be the subset of matrices with determinant = 1. We show that H is a normal subgroup of G directly and by exhibiting H as the kernel of a homomorphism.
Lecture 12
Example of Group: GL(2, R) (3 of 3)
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Example of Group: GL(2, R) (3 of 3)
Abstract Algebra: Let G=GL(2, R) be the group of real invertible 2x2 matrices. We consider two group actions for the group GL(2, R) on itself. We interpret the results in terms of linear algebra and change of basis. We also explain how conjugacy classes of G relate to the diagonalization procedure.
Lecture 13
GT6. Centralizers, Normalizers, and Direct Products
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GT6. Centralizers, Normalizers, and Direct Products
Abstract Algebra: We consider further methods of constructing new groups from old. We consider centralizer and normalizer subgroups, which are useful when the group is non-abelian, and direct products.

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Lecture 14
GT7. The Commutator Subgroup
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GT7. The Commutator Subgroup
Abstract Algebra: We define the commutator subgroup for a group G and the corresponding quotient group, the abelianization of G. The main example is the dihedral group, which splits into two cases.

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Lecture 15
GT8. Group Homomorphisms
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GT8. Group Homomorphisms
Abstract Algebra: We define homomorphism between groups and draw connections to normal subgroups and quotient groups. Precisely the kernel of a homomorphism is a normal subgroup, and we can associate a surjective homomorphism to every normal subgroup.

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Lecture 16
GT9. Group Isomorphisms
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GT9. Group Isomorphisms
Abstract Algebra: In analogy with bijections for sets, we define isomorphisms for groups. We note various properties of group isomorphisms and a method for constructing isomorphisms from onto homomorphisms. We also show that isomorphism is an equivalence relation on the class of groups.

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Lecture 17
Example of Group Isomorphism
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Example of Group Isomorphism
Abstract Algebra: An abelian group G has order p^2, where p is a prime number. Show that G is isomorphic to either a cyclic group of order p^2 or a product of cyclic groups of order p. We emphasize that the isomorphic property usually requires construction of an isomorphism.
Lecture 18
GT10. Examples of Non-Isomorphic Groups
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GT10. Examples of Non-Isomorphic Groups
Abstract Algebra: We note some items that distinguish non-isomorphic groups, including cardinality, orders of elements and subgroups, abelian and cyclic properties, and normality of subgroups. We give a classification of all groups with order less than or equal to 7.

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Lecture 19
GT11. Group Automorphisms
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GT11. Group Automorphisms
Abstract Algebra: We consider the group Aut(G) of automorphisms of G, the isomorphisms from G to itself. We show that the inner automorphisms of G, induced by conjugation, form a normal subgroup Inn(G) of Aut(G), and that Inn(G) is isomorphic to G/Z(G). Examples include Z/3, Z/4, Z, Z/2 x Z/2, and S3.

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Lecture 20
GT11.1. Automorphisms of A4
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GT11.1. Automorphisms of A4
Abstract Algebra: We compute the automorphism group of A4, the alternating group on 4 letters. We have that Aut(G) = S4, the symmetric group on 4 letters, Inn(A4) = A4, and Out(A4)=Z/2. We note that the coset structure splits S4 into even and odd permutations.

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Lecture 21
GT12. Aut(Z/n) and Fermat's Little Theorem
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GT12. Aut(Z/n) and Fermat's Little Theorem
Abstract Algebra: We show that Aut(Z/n) is isomorphic to (Z/n)*, the group of units in Z/n. In turn, we show that the units consist of all m in Z/n with gcd(m,n)=1. Using (Z/n)*, we define the Euler totient function and state and prove Fermat's Little Theorem: if p is a prime, then, for all integers k, p divides k^p - k.

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Lecture 22
GT12.1. Automorphisms of Dihedral Groups
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GT12.1. Automorphisms of Dihedral Groups
Abstract Algebra: We compute Aut(G), Inn(G), and Out(G) when G is a dihedral group D_2n. We also show that Aut(D_2n) always contains a subgroup isomorphic to D_2n and that Aut(D_2n) may be realized as a matrix group with entries n Z/n.
Lecture 23
GT13. Groups of Order 8
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GT13. Groups of Order 8
Abstract Algebra: We classify all groups of order 8 up to isomorphism. There are 3 abelian isomorphism classes and two non-abelian classes, the symmetry group of the square D8 and the quaternion group Q. We define and describe Q and compute Inn(Q) and Out(Q).

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Lecture 24
GT14. Semidirect Products
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GT14. Semidirect Products
Abstract Algebra: Using automorphisms, we define the semidirect product of two groups. We prove the group property and construct various examples, including the dihedral groups. As an application, we show that there exist only two isomorphism classes for groups of order 10 (actually 2p with p an odd prime).

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Lecture 25
GT15. Group Actions
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GT15. Group Actions
Abstract Algebra: Group actions are defined as a formal mechanism that describes symmetries of a set X. A given group action defines an equivalence relation, which in turn yields a partition of X into orbits. Orbits are also described as cosets of the group.

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Lecture 26
GT16. Cayley's Theorem
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GT16. Cayley's Theorem
Abstract Algebra: We consider the left regular action of G on the set X = G. We prove Cayley's Theorem, that every group is isomorphic to a subgroup of a symmetric group, and note a variant when X=G/H. As an application, we show that every group of order p^2 with p prime is abelian.

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Lecture 27
GT16.1 Examples of Cayley's Theorem
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GT16.1 Examples of Cayley's Theorem
Abstract Algebra: We give further examples of Cayley's Theorem and its variant. Examples include the real numbers, the symmetry group of the square, and the quaternion group.
Lecture 28
GT17. Symmetric and Alternating Groups
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GT17. Symmetric and Alternating Groups
Abstract Algebra: We review symmetric and alternating groups. We show that S_n is generated by its 2-cycles and that A_n is generated by its 3-cycles. Applying the latter with the Conjugation Formula, we show that A_5 is simple.

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Lecture 29
Order 12 Subgroups in S5
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Order 12 Subgroups in S5
Abstract Algebra: Find all subgroups in S5, the symmetric group on 5 letters, that are isomorphic to D12, the dihedral group with 12 elements.
Lecture 30
GT17.1. Permutation Matrices
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GT17.1. Permutation Matrices
Abstract Algebra: (Linear Algebra Required) The symmetric group S_n is realized as a matrix group using permutation matrices. That is, S_n is shown to the isomorphic to a subgroup of O(n), the group of nxn real orthogonal matrices. Applying Cayley's Theorem, we show that every finite group is isomorphic to a subgroup of O(n).

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Lecture 31
|Z(G)| for |G|=pq
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|Z(G)| for |G|=pq
Abstract Algebra: Let G be a group of order pq, where p and q are distinct primes. We show that either G is abelian or Z(G) = {e}. We give two proofs: the first uses the class equation, the second uses more elementary methods.
Lecture 32
GT18. Conjugacy and The Class Equation
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GT18. Conjugacy and The Class Equation
Abstract Algebra: We consider the group action of the group G on itself given by conjugation. The orbits, called conjugacy classes, partition the group, and we have the Class Equation when G is finite. We also show that the partition applies to normal subgroups. Finally we apply the class equation to groups of prime power order.

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Lecture 33
Class Equation for Dihedral Group D8
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Class Equation for Dihedral Group D8
Abstract Algebra: Consider the dihedral group with eight elements D8, the symmetries of the square. Find all conjugacy classes of D8, and verify the class equation. Then find all subgroups and determine which ones are normal.
Lecture 34
GT18.1. Class Equation for Dihedral Groups
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GT18.1. Class Equation for Dihedral Groups
Abstract Algebra: We consider the class equation for the dihedral groups D_2n. Conjugacy classes are computed, and we verify the cardinality equation using centralizers. To finish, we consider the partitions for normal subgroups.

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Lecture 35
GT18.2. A_n is Simple (n ge 5)
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GT18.2. A_n is Simple (n ge 5)
Abstract Algebra: Using conjugacy classes, we give a second proof that A5, the alternating group on 5 letters, is simple. We adapt the first proof that A5 is simple to show that An is simple when n is greater than 5. The key step is to show that any normal subgroup with more than the identity contains a 3-cycle.

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Lecture 36
GT19. Cauchy's Theorem
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GT19. Cauchy's Theorem
Abstract Algebra: Cauchy's Theorem states that if p is a prime that divides the order of a finite group G, then there exists an element of order p. We give a proof using the class equation and, as an application, we show that the isomorphism class for a group of order 15 is unique.

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Lecture 37
GT20. Overview of Sylow Theory
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GT20. Overview of Sylow Theory
Abstract Algebra: As an analogue of Cauchy's Theorem for subgroups, we state the three Sylow Theorems for finite groups. Examples include S3 and A4. We also note the analogue to Sylow Theory for p-groups.

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Lecture 38
GT20.1. Sylow Theorems - Proofs
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GT20.1. Sylow Theorems - Proofs
Abstract Algebra: We give proofs of the three Sylow Theorems. Techniques include the class equation and group actions on subgroups.

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Lecture 39
GT20.2 Sylow Theory for Simple 60
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GT20.2 Sylow Theory for Simple 60
Abstract Algebra: Using Sylow theory, we show that any simple, non-abelian group with 60 elements is isomorphic to A_5, the alternating group on 5 letters. As an application, we show that A_5 is isomorphic to the symmetry group of rigid motions of a regular icosahedron.

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Lecture 40
Sylow Theory for Order 12 Groups 1
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Sylow Theory for Order 12 Groups 1
Abstract Algebra: Let G be a finite group of order 12. We apply Sylow theory to study such groups. In Part 1, we consider the abelian cases and A4, the alternating group on 4 letters.
Lecture 41
Sylow Theory for Order 12 Groups 2
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Sylow Theory for Order 12 Groups 2
Abstract Algebra: Let G be a finite group of order 12. Using Sylow Theory, we consider the isomorphism types of G when n_3 = 1 and n_1. In this case, G is isomorphic to either D_12, the symmetry group of a regular hexagon, or a nontrivial semidirect product of Z/3 and Z/4.
Lecture 42
Simple Group 168 - Sylow Theory - Part 1
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Simple Group 168 - Sylow Theory - Part 1
Abstract Algebra: Let G be a simple group of order 168. We calculate the number of Sylow subgroups, number of elements of a given order, and conjugacy class structure. In Part 1, we consider Sylow-p subgroup for p = 3, 7.
Lecture 43
Simple Group 168 - Sylow Theory - Part 2
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Simple Group 168 - Sylow Theory - Part 2
Abstract Algebra: Let G be a simple group of order 168. In Part 2, we compute the number of Sylow-2 subgroups and show that each Sylow-2 subgroup is isomorphic to D_8, the symmetry group of the square.
Lecture 44
GT21. Internal Products
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GT21. Internal Products
Abstract Algebra: We consider conditions for when a group is isomorphic to a direct or semidirect product. Examples include groups of order 45, 21, and cyclic groups Z/mn, where m,n are relatively prime.

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Lecture 45
GT22. The Fundamental Theorem of Finite Abelian Groups
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GT22. The Fundamental Theorem of Finite Abelian Groups
Abstract Algebra: We state and prove the Fundamental Theorem of Finite Abelian Groups. We apply internal direct products to Sylow subgroups in this case. Steps include showing the result for finite abelian p-groups and using the combination rule for cyclic groups with relatively prime order.

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Lecture 46
GT23. Composition and Classification
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GT23. Composition and Classification
Abstract Algebra: We use composition series as another technique for studying finite groups, which leads to the notion of solvable groups and puts the focus on simple groups. From there, we survey the classification of finite simple groups and the Monster group.
II. Ring Theory
Lecture 47
RNT1.1. Definition of Ring
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RNT1.1. Definition of Ring
Ring Theory: We define rings and give many examples. Items under consideration include commutativity and multiplicative inverses. Example include modular integers, square matrices, polynomial rings, quaternions, and adjoins of algebraic and transcendental numbers.
Lecture 48
RNT1.2. Definition of Integral Domain
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RNT1.2. Definition of Integral Domain
Ring Theory: We consider integral domains, which are commutative rings that contain no zero divisors. We show that this property is equivalent to a cancellation law for the ring. Finally we note some basic connections between integral domains and fields.
Lecture 49
RNT1.2.1. Example of Integral Domain
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RNT1.2.1. Example of Integral Domain
Abstract Algebra: Let R be an integral domain. Suppose there exists a y in R such that y + ... + y (n times) = 0. Show that x + ... + x (n times) = 0 for all x in R.
Lecture 50
RNT1.2.2. Order of a Finite Field
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RNT1.2.2. Order of a Finite Field
Abstract Algebra: Let F be a finite field. Prove that F has p^m elements, where p is prime and m gt 0. We note two approaches: one uses the Fundamental Theorem of Finite Abelian Groups, while the other uses linear algebra.
Lecture 51
RNT1.3. Ring Homomorphisms
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RNT1.3. Ring Homomorphisms
Ring Theory: We define ring homomorphisms, ring isomorphisms, and kernels. These will be used to draw an analogue to the connections in group theory between group homomorphisms, normal subgroups, and quotient groups.
Lecture 52
RNT1.4. Ideals and Quotient Rings
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RNT1.4. Ideals and Quotient Rings
Ring Theory: We define ideals in rings as an analogue of normal subgroups in group theory. We give a correspondence between (two-sided) ideals and kernels of homomorphisms using quotient rings. We also state the First Isomorphism Theorem for Rings and give examples.
Lecture 53
RNT1.4.1. Example of Quotient Ring
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RNT1.4.1. Example of Quotient Ring
Abstract Algebra: Are there fields F such that the rings F[x]/(x^2) and F[x]/(x^2-1) are isomorphic? We construct an isomorphism when char F = 2.
Lecture 54
RNT2.1. Maximal Ideals and Fields
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RNT2.1. Maximal Ideals and Fields
Ring Theory: We now consider special types of rings. In this part, we define maximal ideals and explore their relation to fields. In addition, we note three ways to construct fields.
Lecture 55
RNT2.1.1. Finite Fields of Orders 4 and 8
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RNT2.1.1. Finite Fields of Orders 4 and 8
Ring Theory: As an application of maximal ideals and residue fields, we give explicit constructions of fields with 4 and 8 elements. A key step is to find irreducible polynomials (quadratic and cubic).
Lecture 56
RNT2.2. Principal Ideal Domains
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RNT2.2. Principal Ideal Domains
Ring Theory: We define PIDs and UFDs and describe their relationship. Prime and irreducible elements are defined, and conditions for implication are given.

(second version: corrections to definition of prime and irreducible; comment should be 'R UFD implies R[x] UFD, improved proof that, in UFD, irreducible implies prime).
Lecture 57
RNT2.3. Euclidean Domains
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RNT2.3. Euclidean Domains
Ring Theory: We define Euclidean domains as integral domains with a division algorithm. We show that euclidean domains are PIDs and UFDs, and that Euclidean domains allow for the Euclidean algorithm and Bezout's Identity.
Lecture 58
RNT2.3.1. Euclidean Algorithm for Gaussian Integers
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RNT2.3.1. Euclidean Algorithm for Gaussian Integers
Ring Theory: We use the Euclidean algorithm to find the GCD of the Gaussian integers 11+16i and 10+11i. Then we solve for the coefficients in Bezout's identity in this case.
Lecture 59
RNT2.4. Gaussian Primes
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RNT2.4. Gaussian Primes
Ring Theory: As an application of all previous ideas on rings, we determine the primes in the Euclidean domain of Gaussian integers Z[i]. Not only is the answer somewhat elegant, but it contains a beautiful theorem on prime integers due to Fermat. We finish with examples of factorizations in Z[i].
Lecture 60
RNT2.5. Polynomial Rings over Fields
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RNT2.5. Polynomial Rings over Fields
Ring Theory: We show that polynomial rings over fields are Euclidean domains and explore factorization and extension fields using irreducible polynomials. As an application, we show that the units of a finite field form a cyclic group under multiplication.
Lecture 61
RNT2.5.1. Euclidean Algorithm for Z/3[x]
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RNT2.5.1. Euclidean Algorithm for Z/3[x]
Ring Theory: Let f(x)=x^5+2x^3+2x^2 + x+2 and g(x)=x^4+2x^3+2x^2 be polynomials over Z/3. Use the Euclidean algorithm to find gcd(f,g), find the prime factorizations of f and g, and find coefficients for Bezout's Identity in this case. We also find a field in which f(x) factors into linear factors.
Lecture 62
RNT2.6.1. Gauss' Lemma
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RNT2.6.1. Gauss' Lemma
Ring Theory: We consider general polynomial rings over an integral domain. In this part, we show that polynomial rings over integral domains are integral domains, and we prove Gauss' Lemma as a step in showing that polynomial rings over UFDs are UFDs.
Lecture 63
RNT2.6.2. Eisenstein's Criterion
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RNT2.6.2. Eisenstein's Criterion
Ring Theory: Continuing with Gauss' Lemma, we prove Eisenstein's Criterion for Irreducibility and that R UFD implies R[x] UFD. As an example of EC, we show that f(x) = x^4+x^3+x^2+x+1 is irreducible over the integers using substitution.
III. Field Theory
Lecture 64
FIT1.1. Number Fields
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FIT1.1. Number Fields
Field Theory: We give a brief review of some of the main results on fields in basic ring theory and give examples to motivate field theory. Examples include field automorphisms for the rational polynomials x^2-2 and x^3-2.
Lecture 65
FIT1.2. Characteristic p
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FIT1.2. Characteristic p
Field theory: We consider basic properties of finite fields and characteristic p. As in the previous part, we give motivation for Galois theory by considering automorphisms and fixed fields.
Lecture 66
FIT2.1. Field Extensions
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FIT2.1. Field Extensions
Field Theory: Let F be a subfield of the field K. We consider K as a vector space over F and define the degree of K over F as the dimension. We give a degree formula for successive extensions, and consider extensions in terms of bases.
Lecture 67
FIT2.2. Simple Extensions
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FIT2.2. Simple Extensions
Field Theory: We consider the case of simple extensions, where we adjoin a single element to a given field. The cases of transcendental and algebraic arise, depending on whether the kernel of the evaluation map is zero or not. In the algebraic case, we define the minimal polynomial, show it is irreducible, and calculate the degree of the extension.
Lecture 68
FIT2.2.1. Example: Cubic Extension
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FIT2.2.1. Example: Cubic Extension
Field Theory: Let a = 3 + 2^1/3 + 2 * 2^2/3. We show that 1) Q[a] = Q[2^1/3], 2) the minimal polynomial of a over Q is x^3 - 9x^2 + 15x - 25, and 3) a^-1 expressed as a polynomial in a over the rationals. The main technique from linear algebra is the minimal polynomial of a linear transformation.
Lecture 69
FIT2.2.2. Example: Quartic Extension
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FIT2.2.2. Example: Quartic Extension
Field Theory: Let a = sqrt(2) +sqrt(3) in the reals. We show that 1) Q[a]=Q[sqrt(2),sqrt(3)], 2) the minimal polynomial over Q is x^4 - 10x^2 + 1, and 3) an inverse for a^-1 as a polynomial in a over Q.
Lecture 70
FIT2.3.1. Algebraic Numbers
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FIT2.3.1. Algebraic Numbers
Field Theory: We consider the property of algebraic in terms of finite degree, and we define algebraic numbers as those complex numbers that are algebraic over the rationals. Then we give an overview of algebraic numbers with examples.
Lecture 71
FIT2.3.2. Cardinality and Transcendentals
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FIT2.3.2. Cardinality and Transcendentals
Field Theory: We show that the set of algebraic numbers is countable and that any extension of a countable field F by a transcendental is countable. We then give an overview of known results on transcendental numbers.
Lecture 72
FIT2.3.3. Algebraic Extensions
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FIT2.3.3. Algebraic Extensions
Field Theory: We define an algebraic extension of a field F and show that successive algebraic extensions are also algebraic. This gives a useful criterion for checking algberaic elements. We finish with algebraic closures.
Lecture 73
FIT3.1.1. Roots of Polynomials
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FIT3.1.1. Roots of Polynomials
Field Theory: We recall basic factoring results for polynomials from Ring Theory and give a definition of a splitting field. This allows one to consider any irreducible polynomial as a set of roots, and in turn we consider when an irreducible polynomial can have multiple roots. We finish with a definition of separable elements and extensions.
Lecture 74
FIT3.1.2. Roots of Real Polynomials
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FIT3.1.2. Roots of Real Polynomials
Field Theory: We now consider roots of real and complex polynomials. We state and prove the Fundamental Theorem of Algebra, and note its consequences for real polynomials. Then we consider the relation between splitting fields, automorphisms, and roots.
Lecture 75
FIT3.1.3. Example of Splitting Field
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FIT3.1.3. Example of Splitting Field
Field Theory: Let f(x) = x^4 -16x^2 +4. We find the roots of f(x), calculate the splitting field K of f(x) over Q in C, and determine the automorphism group of K.
Lecture 76
FIT3.1.4. Factoring Example: Artin-Schreier Polynomials
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FIT3.1.4. Factoring Example: Artin-Schreier Polynomials
Field Theory: We show that g(x)=x^5-x+1 is irreducible over the rationals using techniques from finite fields. This leads to the definition of an Artin-Schreier polynomial, and in turn we obtain a class of irreducible polynomials over the rationals and prime characteristic.
Lecture 77
FIT3.2.1. Cyclotomic Polynomials
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FIT3.2.1. Cyclotomic Polynomials
Field Theory: We define cyclotomic polynomial as the minimal polynomials of roots of unity over the rationals. We show that the roots of the N-the cyclotomic polynomial are precisely the primitive N-th roots of unity, that the coefficients are integers, and that the degree is phi(N), where phi is the Euler totient function. We also note another formula for cyclotomic polynomials using Mobius inversion.
Lecture 78
FIT3.2.2. Mobius Inversion Formula
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FIT3.2.2. Mobius Inversion Formula
Field Theory/Number Theory: We state and prove the Mobius Inversion Formula. We apply the formula to several examples, including cyclotomic polynomials and the Euler totient function. (Reload: Badly synced audio after compression.)
Lecture 79
FIT4.1. Galois Group of a Polynomial
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FIT4.1. Galois Group of a Polynomial
Field Theory: We define the Galois group of a polynomial g(x) as the group of automorphisms of the splitting field K that fix the base field F pointwise. The Galois group acts faithfully on the set of roots of g(x) and is isomorphic to a subgroup of a symmetric group. We also show that this action is transitive when g(x) is irreducible over F.
Lecture 80
FIT4.2. Automorphisms and Degree
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FIT4.2. Automorphisms and Degree
Field Theory: Having established an estimate on the size of a Galois group of a polynomial, we look at general automorphism groups of finite extensions and give an upper bound on group order by the degree of the extension That is, when [K:F] finite, |Aut(K/F)| is less than or equal to [K:F].
Lecture 81
FIT4.3. Galois Correspondence 1 - Examples
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FIT4.3. Galois Correspondence 1 - Examples
Field Theory: We define Galois extensions and state the Fundamental Theorem of Galois Theory. Proofs are given in the next part; we give examples to illustrate the main ideas.
Lecture 82
FIT4.3.1. Galois Group of Order 8
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FIT4.3.1. Galois Group of Order 8
Field Theory: Let K be Q[sqrt(2), sqrt(3), sqrt(5)], the splitting field of f(x) = (x^2-2)(x^2-3)(x^2-5) over Q. Find the Galois group of K over Q, find all subgroups of the Galois group, and find all subfields of K over Q.
Lecture 83
FIT4.3.2. Example of Galois Group over Finite Field
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FIT4.3.2. Example of Galois Group over Finite Field
Field Theory: We compare the splitting fields of the polynomial f(x)=x^8-1 over the rationals and Z/5. We compute the Galois groups and identify Galois correspondences.